Tuyen tap Phuong trinh bat phuong trinh vo ti huu ti logarit

Noi dung ciia cu6n sach dugc trinh bay thanh ba chuong: Chmmg 1 de cap den phuang trinh bat phuang trinh dang da thuc va huu ty; ChiroTig 2 de cap den phuang trinh, bat phuang trinh v6 t[r]

512.9 PH561T •4 (CB) • ThS PHAN VIET BAC MHAN - CN LE PHUC LUf J f HAT l»HI]liSS •nrnCo ^ D U N G CHO HS GIOI THI TRirOfNG C H U Y E N ^ O N THI THPT QUOC GIA (2trong1) I^NrtA'XUAT BAN • C QUOC GIA HA NOI 5^1 g TS LE XUAN S d N (CB) • ThS PHAN V I E T BAC ThS TRAN NHAN - C N LE PHUC L Q HUUTI VO Tl MO LOGARIT ^ D U N G CHO HS GIOI, THI TRI/flfNG CHUYEN y ^ O N THI THPT QUOC GIA DTR: N H A X U A T B A N D A I H Q C Q U O C G I A H A NQI NHA XUAT BAN DAI HOC QUOC GIA HA NQI 16 Hang Chuoi - Hai Ba Triing - Ha Npi Dien thoai: Bien t$p - Che bkn: (04) 39714896; Quan ly xuat ban: (04) 39728806; long bien tSp: (04) 39715011 Fax: (04) 39729436 * * * Chiu trdch nhi^m xuat ban: Gidm doc - Tong bien tdp: T S P H A M T H I T R A M NHA SACH HONG A N Trinh bay bia: NHA SACH HONG AN ban: Che tdp: Bi^n VAN ANH - PHUONG ANH ^' Doi tdc lien ket xuat ban: NHA SACH HONG A N SACH L I E N K E T PHaONG TRINH - BAT PHUONG TRINH HLJfU TJ, VO TJ, MU, LOGARIT Ma s5: 1L - 99OH2015 In 2.000 cudn, kh6 17 x 24cm tgi COng ti Cd phin VSn h6a VSn Lang Dia chl: S6' Nguy§n Trung Tri;c - P5 - Q Binh Thanh - TP Ho Chi Minh So xua't bSn: 351 - 2015/CXB/4 - 74/DHQGHN, 09/02/2015 Quyg't djnh xuS't ban s6: 120LK-TN/QD - NXBOHQGHN In xong va nOp liAi chieu quy II n3m 2015 Cac em hoc sinh than men! Phuomg trinh, bat phuang trinh la nhOng noi dung can ban chuong trinh toan ph6 thong Co dugc ky nang t6t viec giai phuang trinh, bSt phuang trinh se khong nhung gop phan quan dS hinh va phat trign nang lire giai quyet van de ciia hoc sinh ma giup cac em dat k6t qua t6t nhtjng ky thi quan nhu: thi vao truang Chuyen, thi dai hoc, thi hoc sinhgioi cac d p Vai muc dich ay, chung toi bien soan cuon sach nham cung cap cho ban doc mot he thdng bai tap phong phu, da dang vai nhi§u bai mai la va cac phuang phap giai hieu qua ve phuang trinh, bdt phuang trinh Noi dung ciia cu6n sach dugc trinh bay ba chuong: Chmmg de cap den phuang trinh bat phuang trinh dang da thuc va huu ty; ChiroTig de cap den phuang trinh, bat phuang trinh v6 ty; ChiroTig va Chirong theo thu tu de cap den phuang trinh, bat phuang trinh mu va logarit Trong tung muc, tung phuang phap deu c6 cac vi du minh hoa tieu bieu; c6 phan bai tap de ban doc ren luyen; c6 phin huong din giai bai tap sau de ban doc tham khao, so sanh vai lai giai cua minh Sau moi chuong deu c6 phan bai tap tong hgp, phan Ian la bai tap hay va kho Chung toi hy vgng rang cuon sach "Phuang trinh, bat phiromg trinh va phuang phdp gidi" se thuc su huu ich cho cac em hoc sinh cung nhu cac thay, CO day Toan a truang thong Du da hit sue c6 ging qua trinh bien soan, nhung bg sach kho tranh khoi nhung thieu sot nhat dinh Cac tac gia chan cam an y kien dong gop cua cac thay giao, c6 giao va cac em hoc sinh gan xa de Ian tai ban bg sach se dugc hoan thien hon Mgi y kien dong gop cho tac gia xin quy ban dgc gai ve: nhasachhongan@hotmail.com CAC TAC GIA ChUctng PHLfdNG TRINH, BAT PHtTdNG TRINH HlTU TI §1 TAM THlTC, PHirONG TRINH, BAT PHlTOfNG TRINH BAG HAI 1) DIU cua tarn thipc bac hai Tom tit ly thuyet 7.7 Dinh ly ve ddu cua tarn thuc Cho tarn thuc bac hai f{x)^ax^+bx + c,a^Q Dat A = b^ - 4ac Khi do: NSu A<0 thi af (x)>0 vai moi xeR; NSu A = thi af (x)>0 voi moi x^2a Ngu A>0 thi af{x)>0 vai moi x G ( - O O ; X , ) U ( X ; + C O ) va af(x)<0 vai moi xe{x^;x2), trongdo x, <X2 la hai nghiem ciia / ( x ) 7.2 Dieu kien khong doi ddu cua tarn thuc Cho tam thuc bac hai f{x) = ax^+bx-\-c,a* Dat A = b^ -4ac Khi 2./(x)<0,VxeR« \a<0 l./(x)>0,VxeM<» a>0 A<0 \A<O 4./(x)<0,VxeM<i>| a>0 3./(x)>0,VxeM<:> a>0 A<0 A<0 1.3 Gid tri Ian nhdt, gid tri nhd nhdt cua tam thuc Cho tam thuc bac hai f(x) = ax'^+bx + c,a^ Ta c6 b^^ fix) = a x + — + c, nen I 2aj V 4a b_ Vai a > 0, f{x) c6 gia tri nhoJ nhat la c , dat dugc x = - la 4a Vdfi a < , / ( x ) CO gia t r i Ion nhdt la V i d u Cho cac so thirc x,y,z c- 4a , dat dugc k h i X = - 2a thoa man x + y + z^\ T i m gia t r i lom nhdt c u a b i d u t h u c A = 9xy + I0yz + I z x LcigiaL Thay z = \-x-y vao ^ t a c A = 9xy + \(iyz + nzx^9xy + z(\Qy + \\x)^9xy + (\-x-y){\Qy + \\x) K h a i trien va n i t gon ta c6 ^ = - l l x ' - / + l l x + 10>;-12xv • D o d o \ +{\2y-n)x - -^ =-llx'+(ll-12>;)x-I0/+10>; + \0y^ -\0y + A = De CO gia t r i Ion nhat ciia A t h i phuong trinh c6 nghi?m T a c6 A > < » - / + ; / + 121-44^>0 l l V 495 495 74 22 121 74 ' D o ^ < y- y + — V + < 27 11 11 37 296 148 148 27 25 11 Dku dang thuc xay k h i x = — ; v = — : z = • ^ ^ 74 37 74 495 0V a y gia t r i I o n n h i t ciia A la 148 V I d u Cho hai s6 thuc x,y thay d6i thoa man x^ +y^ = \ T i m gia t r i Ion nhat va nho nhat cua bieu thuc P = • + 2xy + 2y (Di thi dgi hoc khoi B 2008) L&igidL Voi = ta CO = nen P = X 2fx^+6xv) Vai>;^0,datr = - t a c y Dodo P[t'+2t 2(x^+6xy) P^-^ - ^ =- ^ ^ = x'+2xv + / l + 2xy + / + 3) = 2t^+\2tc>{P-2y+2{P-6)t 2/^+12^ • t^+2t + + 3P = , _ ^.l V a i P = 2, phuong trinh c6 nghiem t = — Voi P^2, phuomg trinh c6 nghiem vachi A' = - P ' - P + > o - < P < P = ?> )&h\ = —F=,y = ^ = hoac x = — = , j j ; = - VTo • 3 P = -6 x = — v = — p = hoac x = — i = , y VTI 713 • Vl3 = — = Vn Ket hop lai ta c6 gia tri nho nhdt cua P la - , gia tri Ion nhdt cua P la Vi du Cho a,b^O Tim gia tri nho nhdt cua bi^u thuc b P = a' +b' +^ + - - a a L&igidL Xem P nhu la mot tam thuc bac doi voi bien b Taco P = b' +2b— + -K + +a = b+ 2a la 4a' 4a' + •4a' 4a' D4U bang xay +a + a'>2 -.a' \4a- b= 2a a' = 4a' Vi du Cho cac so duong a, b, c thoa man a + b + c = Chung minh rang a + ab + 2abc< — Laigiau Tugiathiet = - a - c.Tac6 9 a + ab + labc < — • o a + a ( - a - c ) + 2ac(3 - a - c) < — Dat / ( a ) = (2c + l)a^ +(2c^ - c - ) a + ^ > Tachung minh / ( a ) > Ta CO / ( a ) la mot tam thuc bac hai c6 he so ciia a' la 2c +1 > 0, va lai c6 A = ( c ' - c - ) ' - ( c + l ) = ( c - l ) ' ( c ' - c - ) < < c < Tir d o / ( a ) > Dau bang xay a = — ;6 = l:c = — 2 Vi du Cho s6 thuc a,6,c,£/thoa man: + 6^ = 1; c - <i = Chung minh rSng F = ac + bd-cd< ^^^^ (jjsG Nghe An 2005) 2^ L&igidL Tu c-d = taco <i = c - Thay vaotaco + b + 3) {a fa + b + 3^ ( a + b + 3\ -36 c + = ac + (b - c){c F ^ ac + bd - cd = ac + {b - c)d [ J J + K- 3)2= -c^J + (a + b + 3)c - 3b a + + 3V fl + + Y ^ ^ ^ ( a + + 3) 126 c- + Su dung = + 6^ = (a - 6) + 2a6 nen 2(36 = - (a - 6) , thay vao ta c6 (a + + 3)'-126_(fl + 6)'+6(a + 6) + 9-126 - ( a - ) ' + ( a - ) + ll ~ ~ Vi a^+6^=l nen -y[i<a-b<4l Xet ham / ( ^ ) = -f^ + 6? +11 tren [-72;V2] ta c6 / ( r ) < + 6V2 Do F < Dau dang thuc xay d = —, c — , a = — , = 2 Vi dy Cho so thuc a,6,c thoa man di^u kien > 36;abc = 1, Chung minh rang— +6 +c >ab + bc + ca L&igidL— Ta+ c6 b^+c^>ab + bc + ca<^(b + cY-a(b >0 + c) + - <;=>(Z) + c) -2-{b + c) + — + >0 b+c + • 12a > 0, luon dung vi > 36 Vi du Cho hai s6 x,y thoa man -2xy-2x + 4y-7 = Tim gia tri cua x \ihi y dat gia tri Ion nhit (Di thi tuyin sinh THPT Chuyen Qudng Ngai 2013) LcfigidL TsiCO x^+y^-2xy-2x +Ay-1 = Q ti <::>x^ - 2{y + \)x + y^ +Ay-1 = De ton tai gia tri cua x thi phuomg trinh tren phai c6 nghiem, do A' = (>; + 1)^ - - 4;; + > O 2>; < >; < Khi y = A thay vao phuong trinh ta c6 x = Vay X = thi dat gia tri Ion nhdt Bai tap phan 1.1 +z =\ loa man Cho x,y,z thoa man < [2x'+3/+4z'=3 Tim gia tri Ion nhdt, gia tri nho nhat cua y Cho so x,y,z thoa man 1'xxy++y_yz+ z+=zx5= Chungminhrang \<x,y,z< — x^ - xy + y , X +xy + y Cho cac so thuc a,b,c,d cho \ <2 \k a + b + c + d-6 Tim gia tri lom nhat cua P = a^ + b^ +c^ +d^ (Di thi tuyin sinh THPT Chuyen Long An 201 A) Cho a,b la cac s6 thuc thoa man a + b = a^ -ab + b^ Tim gia tri Ion nhat ciia +b^ {De thi tuyin sinh THPT Chuyen Bdc Giang 2007)9 Tim gia tri Ion nhat, gia tri nho nhat cua P= Tim gia tri nho nhat cua A = « X Tim gia tri ion nh4t cua ^ = -4(x^ - x +1) + 3|2x - | , voi - < x < Chobaso x,y,z thoaman • ' ' ^ ' - ^ ' ^ ^ l - ^'^-1 x+y+z^3 Chung minhrSng x^ + y^ + z^ <ll H i r o n g d a n giai b a i tap p h a n 1.1 Ta CO z=1- X- + 3y^ + 4z^ = , do 2x^ + 3>;^ + (1 - X- =3 yf < = > x ^ + / - x - > ; + 8xv + l = , ^ ^ 4> <=>6x^+8(>;-l)x + / - > ; + l = , i^-xviv D I phuong trinh c6 nghiem ta c6 A ' = ( : i ; - ) ' - ( / - ; +1) = - / + > ; +10 > < >^ < Voi y = l ta CO x = z = Vay gia tri lom nhdt cua 3^ la ,, Voi y = —— x = — , z = — Vay gia tri nho nhat cua >> la - ^ 13 13 13 13 2.Tir he ta c6 Do [jj;z = x ^ - x + ' >'z = - x ( > ; + z ) J>' + z = - x j +z =5- x xjr-: ,«f c or - (5 - x ) / + x^ - 5x + = z la hai nghiem cua phuong trinh Phuong trinh c6 nghiem A' > <=> (5 - x)^ - 4(x^ - 5x + 8) > " ' ^ <::>-3x^+10x-7>0<»l<X<- Tuong t u voi y,z ta c6 dieu phai chung minh _ V o i >; = , t a c x ; t O , P = l 'T' ' ' ^^r^^^i ^il^ ml 10 r ^^ X Vai ;;7tO,tac6 P = / \ y r-/ +l y ~ = ^ X , + - +1 -,vai X t= - ^ +^ + J' y Taco y.rs t — f +1 ^=^r^y^<^/'(r'+/ + i) = ^'-^ + i < » ( p - i ) ? ' + ( p + i ) r + ( p - i ) = o De ton tai gia t r i lom nhSt, gia t r i nho nho n h i t cua P, phuang t r i n h n o i tren phai CO nghiem, k h i A'= (P + ) ' - 4(P - ) ' > V a i P = ^ xay k h i r = hay -3P^ + lOP - > <=> ^ < P < = X ^ 'v; u V a i P = xay k h i ^ = - hay x = ->» ^ Vay, gia tri Ian nhSt cua P la 3, gia tri nho nhdt cua P la ^ T a c o \<a<l Tuangtufvai Tudotaco suyra ( a - l ) ( a - ) < < : : > a ' < a - b,c,d P-a^ +6^+c^ + <3(a + + c + ^)-8 = 10 V a i a = 2,6 = 2,c = 1,^/ = hoac cac hoan v i t h i P = 10 Vay gia tri Ian nhdt ciia P la 10 ,2 Tir a + b = a^-ab Dodo {a + bf-4(a Taco DSu +b' =(a _ L L2 + b" =(a + b)^-3ab>(a + b)<0<^0<a + b)[a^-ab + + b^)^{a + bf -^{a + bf =-^(a + bf b<4 + bf <16 bang xay k h i a = = Vay gia tri Ian nh4t cua ct^ + ' la 16 Ta CO A = x'-2oi6x+2 _riV = -2016 rr + 1 Dat ^ = - , ta CO ^ = r - 2016r +1 - 2(r - 504) - 508031 > -508031 11 Do gia tri nho nhk cua A la -508031, dat dugc t = 504 hay x = 504 V.Taco ^ = - ( x ' - ; c + l) + | x - l | = - ( x - l ) ' + | x - l | - = -2x-lp+3|2x-l|-3 Dat/ = |2x-l| v i - l < x < l nen ^e(l;3).Khid6 3^' 3 ' A = -t +3t-3 = - t - - — < — Dau ' =' xay t = — hay x- —, 4 ^ ^ vi -1 < X < Vay gia tri Ion nh^t ciia A la — Dat x = a + l,Z) = j + l,c = z + l Taco a,b,ce[-2;2], a + b + c = Ta can chung minh + 6^ + < Vi s6 s6 a,b,c c6 hai s6 cung d4u, giasu 6,c>0 Khi +b^ <a^ +(b + cf = a' + {-of =2a^<% {b.c^Q , Dau " = " xay \a = -2;6 = 0,c = 2) va cac hoan vi hay a = -2 x = -l;>' = - l ; z = va cac hoan vi TIT ta CO dieu phai chung minh 2) Bit phipcng trinh bac hai Tom tat ly thuyet De xet nghiem cua mot bat phuong trinh bac hai ta dua vao dinh ly vS dSu cua tarn thuc, no phu thuoc dau cua he so a va ddu cua A Gia su /(x) = ay?' + 6x + c, a 0, ls = b^ - 4ac, ki hieu Xj, X2 (xj < X2 ) la hai nghiem cua / ( x ) Khi • Tap nghiem S cua bat phuong trinh / ( x ) > dugc xac dinh boi bang sau a \ — - + + = (x,;x2) S =R S = R\l * - = ( - c o ; x , ) u ( x ; + co) 2al 12 • Tap nghiem S cua bat phuong trinh / ( x ) > ducrc xac dinh boi bang sau a \ - — + S =0 s = -' S = x^;x2 2a + S =R S - (-oo;xi]u[jC2; + co) • Tap ngliiem S cua bat phuong trinh f(x)<0 a — \ • + - = (-oo;x,)u(x2; + oo) S = R\\ • + dugc xac dinh boi bang sau 5= S =0 = (x,;x2) Tap nghiem S cua hk phuong trinh / ( x ) < dugc xac dinh boi bang sau — + - S =R 5=M = (-oo;xi]u[x2; + oo) + 5= s = -' a \ = [x,;x2] 2a Vi du Tim a dk hk phuong trinh sau c6 it nh4t mot nghiem nguyen x^-x + a{l-a)<0 (1) L&igidL Xet A = - 4a(l - a) =4a^ - 4a + = (2a - 1)2 > Va Vay (1) CO hai nghiem Va x = x,; x = ^2 thoa man x, + X j = X X Khi (l) <=> X, < X < X2 Do ' ^ ^ = chi X, nen (1) c6 nghiem nguyen va < < < X2 13 Lai X, + ^2 = nen dieu tuomg duong vai <0<:>a(l-a)<0<=> a<0 a>\' Vay a < hoac a>l Vi dy Tim p de bat phuong trinh sau c6 dung mot nghiem nguyen (2x + p ' - ) ( x - p ' - ) < L&igidL Taco 2x + \4p^-7 Vi -7/?^ + < +^ = 0^x = -lp^ nen bat phuong trinh tuong duong voi -lp'+-<x<p'+- ' ^ •• ' Gia su a la mot nghiem nguyen cua bSt phuong trinh + — > 3, liic bit phuong trinh c6 Neu a < 2, ta CO -Ip^ + — <2 va them nghiem nguyen x = 3, trai voi yeu cm bai toan 15 7, 7, Neu a > 4, ta CO + — > nen p^ >— do -1 p^ + — < —^++ — ao ao -/p+ S — - < nen ^ 4 bat phuong trinh c6 them nghiem la 2; 3, trai voi yeu cku bai toan Neu a = 3, de phuong trinh c6 dung mot nghiem nguyen tuong duong voi < - / ' + - < < p ' + —<4,nentaco—</?'< — 14 14 Vi dy Chung minh rang voi moi x > 1, ta luon c6 <2 L&igidL Ta c6 <2 - • 14 <=>3 X x + - <2 X— x' +1 + -^ ^x'-l^' N2 ^ <2 -1 o3 V ^ y x+— x+— Xy 1^ <2 Xy \2 x^-1 - (vi X > nen o3 >0) X+— x+ Datr = x + - > J x - = , d a u ' = 'xayrakhi x = l,do x > l nen t>2 X V X Bat dang thiic can chung minh tra 3r<2(^'-l)<:>2r-3/-2>0<^(2? + l)(f-2)>0,diingvi t>2 Vi dy Giai bit phuong trinh (x - 6)''+ (x - S)"* < 16 LoigidL Taco (x-6)V(x-8)^ <16 • «((x-7) + l ) V ( ( x - ) - l ) ' < «2(x-7)'+12(x-7)'+2<16«(x-7)'+7(x-7)'-7<0 o - < ( x - ) ^ < l < ^ ( x - ) ^ <l<»-l<x-7<l<:>6<x<8 Vay nghiem ciia bat phuomg trinh la < x < Vi du Tim m cho bat phuomg trinh dung vai moi x e ^ ^ 2x^ +/nx-4 >-6 -^^' ^ x'-x +l Lof gidi Ta CO x^ - x +1 > 0, Vx e R nen bat phuomg trinh tucmg duomg voi x ^ + m x - > - ( x ^ - x + l), V X G R <::>8x^ + ( w - ) x + 2>0, VxeR o A = (m - 6)^ - 64 < -2 < w < 14 Vay-2</n<14 r Bai tap phan 1.2 Tim m de he bat phuong trinh sau c6 nghiem -{m + 2)x + 2m<0 x^ +{m + 7)x + 7m<0 Tim tat ca cac gia tri cua a de bat phuong trinh sau dung voi moi x x+\ —z <1 ax -4x + a-3 Cho f[x) = x^+ax + l voi 3<a< — Giai hit phuong trinh f{f{x))>x Tim m de bat phuong trinh sau dung voi Vx e R (m-l)x^ +{m + \)x + m-\>0 Tim m de bat phuong trinh sau c6 tap nghiem la M ^^'^ ^ 3x^ -mx + ^ 1<— <6 lx^-x + \ Huang dan giai bai tap phan 1.2 x ' - ( w + 2)x + 2m<0 (1) Ki hieu + ( w + 7)x + 7m<0 (2) Taco A, = ( w - ) ^ > ; A = ( m - ) ^ > Ta CO phuong trinh Phuong trinh x^ + - (m + 2) x + 2w = luon c6 nghiem x = m hoac x = + 7) x + 7m = luon c6 nghiem x = -m hoac x = - Voi m = hoac voi m = thi he v6 nghiem Voi m^2\m^l, m>0 nghiem cua ( l ) la 5, =(w;2) hoac S^-[2•,m)•, 'nghiem cua (2) la ^2 = ( - m ; - ) hoac ^3 = ( - ; - w ) Ro rang 5, f l ^ j = nen he v6 nghiem Voi m<0 nghiem cua ( l ) la 5, = ( w ; ) , nghiem cua (2) la -{-l;-m) 16 Ta CO iS", n <S'2 ?t nen he luon c6 nghiem Vay w < 2, Ta tim diSu kien dS ox^ - 4x + a - 9^ 0, Vx e R Taco ca^-4x + a-3^0yxsR^ A' = - a ( a - ) < <=> a<-l a>4 Neu a<-l thi ox^ - x + a - < , V x e M Tirdo x+1 <l,Vx( ax - x + a - <=>x + l>ax^ - 4x + a - , V x e M <=>ax^-5x + a - < , V x e M <=>A = - a ( a - ) < (vi a<-l) \a<-l - V 4-V4T <=>S , <=>a< [4a'-16a-25>0 N6u a > thi ox^ - 4x + a - > 0, Vx e M Tirdo :0 x+1 •<l,VxeR ox - x + a - o X +1 < ax^ - 4x + a - 3, Vx e M oox^ - 5x + a - > , V x e M O A = - a ( a - ) < ( v i a>4) <=> + V4T a>4 4a'-16fl-25>0 Ket hop lai a e -Qo; 4-741 u 4+V4T -; + oo Taco / ( / ( x ) ) >x<=>/(/(x))-x>0 Taco /(/W)-^ = /(/W)-/W+/W-^ i / ( x ) + - (xL+.flx+-i-W-/fx)-x-~':":7:^7,'i '/'{x)-x'] ^(x^ +(a-\)x + a[f{x)-x] =[f{x)-x][f{x) + x + a] el + \)[x' +{a + \)x + a + 2) Bat g{x) = x'+{a-l)x + lc6 A^=a'-2a-3 h(x) = x^+{a + \)x + a + CO A^=a^-2a-7 Do < a < | nen tudo ^ ^ , d o d h{x) = x^ +{a+ \)x + a + 2>0,\fxeR, f(f{x))-x>0 <=>g(x)>0<=>x^ + ( a - l ) x + l > O o l-a-Va'-2a-3 x> l-a + V a ' - a - fm-l>0 4.Tac6 ( w - l ) x ^ +(m + l)x + m-l>0,\/xGR<^ A<0 m-l>0 /w-l>0 / w l > m<•ow>3 (m + 1)2 - ( w - l ) <0 [-3m'+10/n-3<0 m>3 Ta CO 2x^ - X +1 > 0, Vx G M Do bat phirong trinh tuong duong voi 3x^ - mx + -,Vxe: 1< x^ + ( l - m ) x + 4>0,VxGlS 2x'-x + l 9x'+(m-6)x + l>0,Vxe 3x -tnx + < , V x e : [ 2x'-x4-l A,=(l-m)'-16<0 {-3<m<5 A,=(m-6)'-36<0 [0<m<12 Vay vdd < m < thi bdt phuong trinh c6 tap nghiem la R 18 3) Mot s6 dang phu'O'ng trinh hCeu t i dipa v§ b^ic hai Dang t6ng quat: au^ (x) + bu{x) + c = 3.1 Dang trung phuong ox^ + + c = Cdch gidi: Dat = ^ > dk dua wh phuong trinh bac hai theo t 3.2 Dang (x + a)"^ + (x + Z?)'* = c Cdch gidi: Dat x + = t se thu dugc phuong trinh trung phuong theo / 3.3 Dangnghich dao ox'* + bx^ + cx^ ±bx + a ^ 0, a ^0 Cdch gidi: Chia ca hai ve cua phuong trinh cho n x^ 9^ thu dugc phuong trinh + b x± a = / (hoac x- \ — -t), bieu dien x^ + theo t, thay vao phuong Dat X + — V X X X trinh tren se thu dugc phuong trinh bac hai theo t 3.4 Dang h6i quy ax"^ + bx^ + cx'^ + dx + e = 0, voi — = Kb a Cdch gidi: Gia su — = k^0 Chia ca hai vg cua phuong trinh cho x^ ^0 thu b dugc phuong trinh aa x ^ + ^ + x + — + c = I xj k ~ -) k~ Dat x + —= bieu dien x + ^ theo t, thay vao phuong trinh tren se thu X x^ dugc phuong trinh bac hai theo t 3.5 Dang (x + a)(x +fe)(x+ c)(x + c/) = e, voi a + <i = + c CflcA ^ifl/; Viet phuong trinh duoi dang ((x + a)(x + ^))((x + 6)(x + c)) = e «> ^x^ +{a + d)x + ad^{x^ +{b + c)x + bc^ = e D?lt x^ +(a + d)x = r thu dugc phuong trinh bac hai theo / 19 3.6 Dang (x + a)(x + b)(x + c)(x + d)-ex ,\(n ad = be Cdch giai: Viet phuong trinh duai dang ((x + aXx + cl)){(x + b){x + c)) = ex^ <=>(x^ + ( a + (i)x + fl<ij(x^ +(b + c)x + bc^^ex^ Cdch Dat x^ + a + b-\-c + d x + ad-t thu dugc phuong trinh bac hai theo t Cdch Xet X = 0, thu true tiSp Xet X ?t 0, chia ca hai ve cho x^ ta c6 ad x + {a + d) + — X+ be (6 + c ) + -e ad Dat / = X + — , dua ve phuong trinh bac doi voi t \2 ax = b, a^O,x^-a X 3J Dang x^ + yx + aj Cdch giai: Viet phuong trinh duai dang X- y ax + 2a = b^ x+a x+a - + 2a (x + aY x+a X Dat x+a = ?, thu dugc phuong trinh bac hai theo t ax 3.8 Dang -^ + X +inx + k bx ^ = e,abc ^ X +nx + k Cdch giai: Xet x = Khong phai la nghiem ciia phuong trinh Xet X ^ Vidt phuong trinh dudi dang —— + —— = c x+m +— x+n+— X X Dat x + — = t, phuong trinh tra — ^ + — ^ = c, tu dua X t+m t+n phuong trinh bac hai theo 3.9 Dang dang cap bac hai aw^(x) + ZJV^(x) + CM(X)V(X) = Cdch gidi: Gia sii v(x) ^ Chia ca hai vg cho v^(x) ^ 0, r6i dat = f d6 v(x) dugc phucmg trinh bac hai theo t 3.10 Dang au^(x)v^(x) + b{u{x) + v{x)f +c = 0, vai u(x)-v(x) = k Cdch gidi: Viet phuong trinh da cho ve dang au^ (x)v^ (x) + b {u{x) - v(x))^ + M ( X ) V ( X ) + C = aw^ (x)v^ (x) + 46w(x)v(x) + c + 6yt^ = Dat u{x)v(x) -1, thu dugc phuong trinh bac hai theo t 3.11 Dang x'=ax^+bx + c Cdch gidi: Ggi a la s6 thuc thoa man 6^ = 4(a + 2a)(c + «^) (Day la mot phucmg trinh bac ba d6i vai a nen luon ton tai a) Khido: x'=ax'+Z>x + c ^x' +2ax' ={a + 2a)x^ +bx + (c + a') j <^(^x^ +af ={a + 2a)x^ +bx + (^c + a^y NSu a + 2a ;t 0, vl phai (a + 2a)x^ + 6x + (c + ) la mot tam thuc bac hai CO A = b^-4{a + 2a)[c + a^) = Q N6u fl + 2a > 0, ta CO (JC^ + a)^ = |Va + 2a x + Vc + o^j x^ +a = |Va + 2a x + Vc + o^ j x^ + a =-{^4a + 2a X + 4c'+a^ NIU <3 + 2a < thi FP < 0, > 0, do ding thuc khong xay NSu fl + 2a = 0=>6 = 0.Tac6: (x^ + a)^ =(c + a^), tu tim dugc x 12x 3x Vi du Giai phucmg trinh—^ z = ^ ^ x'+4x + x'+2x + {Dk thi tuyin sink THPT Chuyen DH Vinh 2010) 21 LoigidL DiSukien: Ro rang x = +Ax + 2^Q, +lx + 2^Q khong phai la nghiem cua phuang trinh D o p h u o n g trinh tuomg duong v a i 12 X + - + X + - X + -1 X D5t X + - = ? phuang trinh t r a X 12 t+A =1 t+2 <:>9/ + 12 = (/ + 4)(r + 2) o r - / - = 0<» t = -\ t =4 V a i f = - l t a c x + — = - l < = > x ^ + x + = 0, v n g h i e m X Vai t = t a c x + - = 4<::>x^-4x + = < ^ x = 2±V2 X Vay nghiem ciia phuang trinh l a x = ± V2 V i d u Giai phuang trinh x " - 3x^ - 2x^ + 6x + = (De thi tuyin sink THPT LM Chuyen DH Vinh 2009) gidL V i X = khong la nghiem cua phuang trinh nen phuang trinh tuang duong v a i x'+4-3x + - - = ^ X X X 2^ -3 -— x) V X - 2' + = V Dat ? = X - - , k h i phuang trinh c6 dang - 3t + = 0<^ X Vai/ = 2<»x — X =2«x^-2x-2 =0 « x = t =2 t = \' l±V3 X Wai t = l<^x = l<i:>x^ -x-2 = x =- l 0^ x =2 Vay nghiem cua phuang t r i n h la x = ± V3 , x = - l ; x = • - { V i d u Giai phuang trinh (x^ + x + l)(x^ + x + 2) = 12 (Di thi tuyin sink THPT Chuyen DH Vinh 2008) 22 L&igidL Qat +x + l = t,(t >0) phuang trinh da cho tra thanh: /(r +1) = 12 » r^+^-12 = 0»r = , d o r > Vol/ = taco +x + l = <=> + x - = <=>''x = l x^-2 Vay nghiem cua phuang trinh la x = -2; x = 1, Vi du Giai phuang trinh jc" + - 8JC -12 = (Di thi tuyin sinh THPT Chuyen DH Vinh 2006) LoigiaL Ta c6 / + 4x' - 8JC -12 = <=> / + + 4x' - 4JC' - 8x -12 = o (x' + 2x)' -4(x' + 2x)-12 = «-4r-12 = 0, vai f = x' + 2x > -1 t^-2 <^t = 6(t>-\) t=6 <;:^x'+2x-6 = 0«x = - l + V7 Vay nghiem ciia phuang trinh la x = -1 ± V? Vi du Giai phuang trinh x^ + 25x^ = 11 (x + 5)^ (De tuyin sinh THPT Chuyen Lam Son Thanh Hoa 2013) L&igidL *) Dieu kien x^-5 Phuang trinh da cho tuomg duang vai 5x + -lOx' X x + = 11<:> x + 5^ + 10-x + -11 = V Dat t = x+5 , / x+s; ,tac6 r + / - l l = o N2 t^l t^-n x^ - X - = x^ +1 Ix + 55 = 0(v6 nghiem ) x+5 =1 = -11 x+5 I + V2T ox = m 1±V2T Vay phucmg trinh c6 hai nghiem x = Vidu Giai phucmg trinh (x^+3x +3)\ +(x^+3x + 5) =82 (Di tuyin sink THPT Chuyen Lam San Thanh H6a 2013) LM giaL Dat ? = (^-1)'+ + 3x + Ta c6 =82<»r'-4f'+6/'-4/ + l + f ' + / ' + r ' + ^ + = 82 t = -2 r = <=> + - 40 = o t =2 r =-10 Vai t = -2 taco +3x + = -2<»x'+3x + = (v6nghi?m) Vcfi / = taco x^ +3x + = 2<»x^ +3x + = o x = - l x = -2' Vay phucmg trinh c6 hai nghiem x = - ; x = - Vi du Bilt phucmg trinh x" + ox^ + 6x^ + ax +1 = c6 nghiem Chung minh rang a +b >- L&igidL (Di thi tuyin sinh THPT Chuyen Ha Tinh 2014) Gia su x^ la nghiem ciia phucmg trinh da cho, ta c6 Ro rang Xg ^ Ta c6 XQ + OXQ + 6X0 + OXQ + = Xn + OXn + ZJXQ + OXQ + = <=> XQ + OXQ + + — + ^ = ^0 ^0 \2 Xo+- +a Xo+- + - = Dat r = Xn + — ta CO t\>2 Tu phucmg trinh tucmg duong t^+at + b-2 = 0<:^at + b = 2-t^ 24 Do ( - f = (at + bf <{a'+b'){t'+!)<:> — Tachimg minh — — a'+b'> +4 > - <=> 5t^ - 24t^ + \6 > <^ [St^ - ) ( / ^ - ) > , luon dung v i \t\>2 Vay ta c6 di6u phai chung minh V i du Chung minh rang phuomg trinh sau khong c6 nghi?m nguyen (x-y)(x-2y)(x-3y)(x-4y) + /+2 = z\ (De thi tuyin sink THPT Chuyen Phan Boi Chdu Nghe An 20\4) L&igiaL Ta c6 (x-y)(x-2y){x-3y)(x-4y) <=> ((X -y){x^[x^-5yx Bat t = x^-5xy + 4y)){{x + y' +2 = z^ - 2y){x - 3y)) + /+2 + 4y^)[x^-5yx + 6y^) + = z' y*+2^z\ 5y\Tac6 (t'-/)(t'+/) + /+2 = z'<^t'+2 = z'<=>(z-t')(z Khong m4t tinh tong quat gia su z > 0, ta c6 • ^ ' ^ ^ + t') = ^, z-e=\ suy z = — V6 ly Vay phuong trinh da cho Ichong c6 nghiem nguyen V i du Giai phuong trinh = ~x^ - 4x + L&igidi Taco x' =-x' -4x + ^x' 'x'+l = +2x^ +l^x' x-2 4-l = - ( x - ) -4x + c^[x^ +\f - x + = (vonghiem) ={x-2) -1±V5 <=>x = x'+x-l =0 Vay nghiem ciia phuomg trinh la x = -1±V5 25 Bai tap phan 1.3 Giai phuang trinh x'+4 -5x = x'-2 Giai phuong trinh x" = +1 OJC + Giai phuang trinh x" - 4>/3x - = Giai phuang trinh x" = 2x^ + 2x — Giai phuang trinh (x^ -2x)^ + ( x - l ) = x ( x - l ) - "-^ (Di thi tuyin sinh2 THPT Chuyen Quang Trung Binh Phuac 2010) ^x^-P + Giai phuang trinh - X- - U - j = ( D J thi tuyin sinh THPT Chuyen Lam Som Thanh H6a 2014) rx-n rx+n Giai phuang trinh(x + 2)(x + 3)(x + 8)(x +12) = 4x^ Giai phuang trinh (4x + l)(l2x - l)(3x + 2)(x +1) = Giai phuang trinh x" - 2x^ - 4x^ + 4x + = 10 Giai phuang trinh (x - 6)' + (x - 8)^ = 16 Huong dan giai bai tap phan 1.3 Dilu kien: x ^ - v t « x ^ ±V2 Taco x ' + 4^ - x = 0«x'+4-5x(x'-2) = 0<=>x'-5x'+10x + = Vai X = 0, khong phai la nghiem ^ •^ Voi X ^ 0, chia ca hai \k cua phuang trinh cho x^ ta c6 2^ + = x^-5x + — + 4- = 0<:> X X Xv Xj - r=l — \ X- — x) Dat ? = x - - , t a c /^-5? + = o t = A X 26 x= -\ Vcri/ = l , t a c x - - = l O A ; ^ - x - = < ^ x=2 Vai / = , t a c x - - = 4<:^x^ - x - = 0<::>2±V6 Vay nghiem ciia phuomg trinh l a x = - l ; x = 2;x = ± V Taco X* = 3x^ + \0x + <:>x' + 2x^ +\ 5x' +10X + <:^[x' + if =5{x + if x ' + l - V ( x + l) x'-V5x + (l-V5) = x'+l x ' + V x + (l + V5) = = -yf5(x + l) Taco x ' + V x + (l + V5) = v6 nghiem Vay nghiem ciia phuong trinh la x, ^ = ^ Taco x'-4yf3x-5^0^x' <=> - + =4yf3x + + x ' + = x ' + 4>/3x + o ( x ' + ) ' = (x + N/3 ) ' x ' + l = - V ( x + V3) Ta CO x ' = 2x' + 2x - - o x ' + 2x' + = 4x' + 2x + 4 \ x'+l-2x + - 2x.i 2J x^+l = - 2X.1 x'-2x + - - x^+2x + - = V2- Vay nghiem ciia phuong trinh la x = ± y Phuong trinh tuong duong voi (x^ - 2x)^ - 2(x^ - 2x) - = Dat ? = x ' - x T a c r ' - r - = 0« t = -l t=3 Voi / = - , ta CO x^ - 2x = -1 <:> X = Voi ? = 3,tac6 x^-2x = o x ^ - x - = 0<:^ x = - l x=3 Vay nghiem ciia phuong trinh la x = -1; x = 1; x = X —1 X +1 Dieukien x^±2 Dat fl = ^ ^ ; = ^ ^ x+2 x - 2= 0^ a^b TaCO -4ab + 3b^ ^Oo{a-b){a-3b) a = 3b x-1 x + Voi a = b<^ x + x - <»(x-l)(x-2) = (x + l)(x + 2)<»x x-1 ^x+1 Voi a = 36 = 3- <»(x-l)(x-2) = 3(x + l)(x + 2) x+2 x-2 «2x'+12x + = o x = -3±V7 Vay nghiem cua phuong trinh la x = 0; x = -3 ± Taco (x + 2)(x + 3)(x + 8)(x + 12) = 4x' (x + 2)(x + 12)J[(x + 3)(x + 8) -4x^ «(x' + 14x + 24)(x^ +1 Ix + 24) = 4x.2 , 25 x + 24,tac6 Dat t = x'.2 + — 28 t = -x f ( ^ =\x^<^e=—x^<^ t + -x t5 [ )V ) t = —x x = -4 25 x^ + — jc + 24 = - x < ^ x ^ +10x + 24 = 0<^ x = -6 Voi t = -x,tac6 2' Vai — X , taco x'+—x + 24 = -^x<^x'+l5x + 24^0<^x=~^^-'^ Vay phuong trinh c6 cac nghiem x = -4;x = -6;x- -15±Vl29 Cach khdc: Voi X = 0, khong phai la nghiem Voi X , tir phuong trinh (x^ +\4x + 24)(x^ + l l x + 24) = 4x^ chia cahai v l cho x^ ta CO ^x'+14x + Y x^+11x4-24^ 24 , = o x + — + 14 x + — + 11 = X X 24 Dat ^ = X + — , quy ve phuong trinh bac doi voi t X Taco (4x + l ) ( l x - l ) ( x + 2)(x + l) = ^- <::> [(4x + l)(3x + 2)][(l2x - l)(x + ) ] = <t:> ( l x ' + Ix + ) ( l x ' + Ix - ) = Dat r = 12x^ +11X + - Taco ( t + -A I jV 3^ t- 2 , 25 r= - • ^~ 29 Vod ^ = - t a c 12x'+llx + - = -<=>12x'+llx-2 = o x = - ^ ^ ^ ^ ^ 2 24 taco 12x^+llx + ^ = -|<::>12x^+llx + = 0(v6nghi?m) Voi ' = - I I + V217 V§y nghiem cua phuong trinh la x = 24 Voi X = 0, khong phai la nghiem cua phuong trinh Voi X 5t 0, chia ca hai v6 cua phuong trinh cho x^ ta c6 x'-2x-4 +- + = 0ox'+4-2 X — -4 =0 X Dat / = X — Ta X Voi ^ = , t a c o CO X - 4r X X 2/ = <=> t = 2' — = 0<»x^-2 = 0«x = +V2 X Vdir = , t a c x — = o x ^ - x - = 0<::>x = l + V3 X Vay nghiem cua phuong trinh la x = +V2 ,x = \±^ 10 Dat r = x - T a c (f + l ) ' ' + ( / - ) ' = < ^ / ' + f ' + = 16<»f'+6r'-7 = 0<=>f'=l t>0) x=6 ^^T^4 Taco / ' = l o f = l x = 8' Vay nghiem cua phuong trinh la x = 6; x = (vi .8 30 §2 PHlTOfNG T R I N H , B A T PHlTONG T R I N H B A G C A O V A PHlTOfNG T R I N H , B A T PHlTONG T R I N H HCTU TI 1) Mot s6 k i t qua v a nhan xet 1.1 Dinhli Viet Gia su P{x) = aQx" +a^x"''^ •¥ +a^_^x + a„,{aQ^Q,n>2) c6 cac nghiem la n Dat ^2 = 0-, = 1=1 X ^/^y = XiX2 X„ \<i<j<n Khi 0> = ( - ) * ^ 1.2 Dinh ly Bezout Cho P(x) = aQx" ^ ^ ^ j ^ y;^ + + a„_,x + a„,('3o ^ , « > 1) va XQ ^ = K h i XQ la nghiem ciia P(x) va chi P{x) = (x - Xo)S(x), Q{x) la da thucbac n-l 1.3 Lucre Hooener Cho P(x) = a o ^ ' ' + a i x ' ' " ' + + a„_,x + a „ , ( a o ' ^ , n > l ) va X g e M K h i ton tai nhat bo so thuc bQ,bi, b„ saocho Pix) = (x-x,)(b,x"-' I +b,x"-^ + + b„_2X + b„_,) + b„ -.Am Bo so 6o,6j, 6„ dugc xac dinh bai 6o = ^c^* - ^o^*-i +cii^,k = 1,2, ,« Chu y rang Z?„ = P(XQ), nen XQ la nghiem cua P(x) va chi Z>„ = 1.4 Dinh ly ve nghi?m hihi ti Gia su s6 huu t i r = —(p e Z , o eN*,(p,flr) = l ) la nghiem ciia da thuc he s6 nguyen P{x)^aQx"-^a^x"'^+ + a„_^x->ra„,{aQ^Q,n>\) K h i p\a„ He qua: a) M o i nghiem huu ti cua da thiic vai he s6 nguyen P ( x ) = x" + + + <3„_,x + a„ deu la so nguyen b) Moi nghiem nguyen ciia da thuc he s6 nguyen P(x) = ao^" +a,x""' + + a„_,x + a„ deu lauac ciia a „ 7.5 Dinh ly ton tai nghiem thuc a) Moi da thuc bac le deu c6 nghiem thuc b) Gia su a,b{a<b) P{a).P{b) la hai s6 thuc va P(x) < Khi P ( x ) c6 nghiem thuoc khoang la da thuc thoa man (a;b) 1.6 Dinh ly phdn tich da thicc he so thtrc Moi da thuc P{x) = aQx" ^-a^x"'^+ dang P{x) = + a^_^x + a^ dSu phan tich ducrc flo(^-)-(^-h)[^^ + + ) - ( ^ ^ +PiX + qi), a day pf -4^, <0,k + 2l = n Noi each khac, moi da thuc dku phan tich dugc cac thua s6 bac nhkl, bac hai ,r v He qua: a) Phuong trinh bac n c6 nhiSu nh4t n nghiem b) Phuong trinh a^x" + a , x " " ' + +fl„_,x+ a„ - 0,n e N c6 nhigu hon n nghiem va chi aQ =^ a^ = = a„ = 1.7 Dinh ly ve ddu cua da thirc Gia su a va Xj < X j < < x „ Khi da thuc i ' ( x ) = a(x - XjXx - X2) (x - x„) khong doi dau m6i khoang (x,;x,_^,),/ = , l , n , vai XQ = = +°o va hai khoang ke bat k i P ( x ) c6 dau nguac .5 Vi du Chung minh rang phuong trinh x + x - = c6 dung nghiem L&i gi&L Dat P ( x ) = x ' + x - l Ta c6 c6 P ( l ) P ( - l ) = l ( - ) < nen phuong trinh da cho c6 nghiem x^ e ( - ; l ) Matkhac v o i m o i x < X Q , P{x) = P{x) = x^ +x-\>xl+Xo-l + x-I < xl + -1 = vayai moi X > X Q , = 0, nen phuang trinh da cho Idiong c6 nghiem tren moi khoang ( - 0 ; XQ ) va (XQ ; + 0 ) T u suy diSu phai chung minh Vi du Cho da thuc P{x) = x' +x'-9x' +ax^ +bx + c BigtrSng P(x) chiahStcho ( x - ) ( x + 2)(x + 3) Hay tim P(x) L&igidL Taco P{x) chiahgt cho ( x - ) ( x + ) ( x + 3) nen P(2) = 0; P(-2) = 0; P ( - ) = T u thay vao ta c6 a = - ; = 20; c = - Da thuc can tim la P(x) = x^ + x^ - 9x^ - x^ + 20x - ^ Vi du Tim m de phuang trinh ~x ^ - x + m + - = 0c6 nghiem phan biet CO tong binh phuang cac nghiem Ian hom 15 '- L&igidL T a c o - x ^ - m x ^ - x + m + —= (1) c ^ ( x - l ) x ^ + ( l - m ) x - - m =0 x =l g{x) - x^ + (1 - 3m)x - - 3m = ,5 ; v / De phuang trinh (1) c6 nghiem phan biet thoa man Xj + X2 + X3 > 15 va chi Idii g(x) = CO nghiem X j , X2 phan biet khac va thoa man x,^>14 <=>|An| > Vi du Tim m de phuang trinh x^ - 3mx^ + 9A: - = (1) CO nghiem phan biet lap cdp s6 cong L&i gidL Gia su phuang trinh c6 nghiem Xj; X2; JC3 Ta c6 Xj + X j + = 3m De X j ; x ; JC3 lap cap so cong thi X2 = m la nghiem cua phuang trinh (1) m Ta CO -2m^ + 9m - = « m=l m = -1 + Vl5 -1-Vl5 _Thu, ,lai ta CO, m = - l - > / l thoa , man ^ Vay m = -l-4l5 Vi dii Tim m dk phuong trinh - 3mx^ - (m + l)x - = c6 nghiem phan biet lap cip s6 nhan L&i gidL Gia su phuong trinh c6 nghiem phan biet x^;x2;x^ Ian luot lap cdp so nhan Khi ta c6: gix) = {x-x^){x - ){X - X ^ ) jCj + ^2 + JC3 = Suy ra: Vi X^X2 x^x.^ = ^2 + X2XT ^ 3m + X^X-^= =2^ ^2 -m -1 = nen ta c6: - m - = + \/2.3m <=> m = — 3</2+l Voi m = —3 — ^ + ,1 thay vao tinh nghiem thiy thoa man Vay m = - ^ ^ + 1" Bai tap phan Bai Tim m de phuong trinh x'^ - (3m + 2)x^ + 3m +1 = c6 nghiem phan biet deu nho hon Bai Tim m de phuong trinh x^ - 2(m + \)x^ + 2m +1 = c6 nghiem phan biet lap cap so cong > , Bai Cho 2a + 66 +19c = Chung minh rang phuong trinh ox^ + 6x + c = luon CO nghiem tren 34 Bai Cho P{x) la da thiic xac dinh tren [a; b] va n dilm x^, , ,x„ e [a;6] Chiing minh r&ig t6n tai c e [a; b] cho P{c) = — [ ) + P(x^) + + n Bai Cho da thuc vai he so thirc ^ ) > P ( x ) = x"+a„_ix""'+ + a,jc + ao, Q{x) = x^ +X + Bik ring da thiic P{x) c6 « nghiem thuc phan biet va da thuc khong CO nghiem thuc Chung minh ring P(2)> — Huang dan giai bai tap phan l.Taco jc'*-(3m + 2)x^+3m + l = <^[x^ - \\[x^ - Zm-\ = « x^^\ jc^=3m + l(2) Tir de phuong trinh da cho c6 nghiem phan biet deu nho hon va chi (2) CO hai nghiem phan biet khac ±1 va nho hem 0<3m + l < f <=> r3 \ 3/n + l?il Xet phuong trinh x^ - 2{m + l)x^ + 2m +1 = (1) Dat t^x^,t>0 thi (1) tro thanh: /(t) = - 2(m +1)? + 2m +1 = Dd (1) CO nghiem phan biet phai c6 nghiem duong phan biet A' = m^>0 <=> < = 2(m + l)>0«<! m> — P = 2m +1 > m^O Goi < ?2 la nghiem cua /(O = 0, cac nghiem cua (1) Idn lugt la: De x^,x2,x^, lap cap so cong - x^ - x^ - X2 = x^ - x^ <=> t2 x^ <^ X2 - it Dat Ta CO m = 9(m + l - m j < : : > m = 4(m + l ) < » m= 4 (thoa man) Vay m-4;m m=— * v3 n CO g(x,) = 5m = 4m+ -5m = 4m + — = ax^ + Z?x + c ta c6 P{x) la da thuc xac dinh tren R P(0) = c,18P - PCX,) - - „Do = 2a + 66 + 18c = - c phuomg trinh P(x) - luon c6 nghiem tren 1r Dat g(x) = P(x) — [P{x,) Ta + P(x,) )+ „P{0)P , „ „ r p <0 nen »4 + + P(x„)], g{x) cung la mot da thirc ) + + P(x„) n gix,) = P{x,) [Pix,) g{x„ )^P{x„) n + P(x,) [P{x,) + + + P{x,) + + P(x„) P{x„) Do g(x,) + g ( x ) + + g(x„) = 0, do ton tai Z,; e {l,2, ,«} cho g(x,)<0; Ta c6 g{Xj)>0 < , do t6n tai c nSm giua x,\Xj (khi hien nhien c e [a; b\) cho g(c) = 0, ta c6 P{c) Do P{x) = -[P{x,) + P{x,) + + P{x„)\ CO « nghiem thirc phan biet nen P{x) = (x - X, ) ( x - ^2 ) (x - x „ ) Do ^ ( ^ ( x ) ) khong c6 nghiem thirc nen g ( x ) ^ x,,Vx e M , ( / = 1,2, ,«) <;:^x'+X + - X , 9^0,Vxe]R 36 o A < <=> < - X,, (/ = , , n ) , Dod6P(2) = ( - x , ) ( - x , ) ( - x „ ) > i , 2) Phiro'ng phap giai 2.1 Phuang phdp phan tich Vi du Giai cac.phuang trinh sau 1) x^- 8x^+21x-18 = 2) x^- X - V = 3) (x - i f + (2x + 3)^ = 27x^ + 4) x^-3^^x^ + 7x-^/3=0 5) (x + l)'(x + 2) + ( x - l ) ' ( x - ) = 12 Lei giai 1) Nhan thay phuang trinh c6 cac nghiem x = 2, x = Tu ta phan tich nhan tii , x^- 8x^+21x-18 = 0<:^(x-2)(x^-6x + 9) = o 2) Ta nhSm x-yjl ^x = x=3 la nghiem Khi do, phan tich dugc: x ' - x - V = ( x - V ) ( x ' + V x + l) = "x-V2=0 x^ + V2x +1 = (v6 nghifm ) <»x = V2 ' ' • ii'y- \ 3) D I thdy n§u su dung hang ding thuc + d6i vai ca hai vg ta thSy xu4t hien nhan tu chung la 3x + ( x - l ) ' + ( x + 3)'=(3x)'+2' o (3x + 2)[(x - ) ' - (x - l)(2x + 3) + (2x + 3)'] = (3x + ) ( x ' - 6x + 4) 37 o (3x + 2)[9x^ - 21x-18) = o 3x + = 9x'-21x-18-0 X= — x = -3 x= — 4) Ta CO X = V3 la nghiem cua phuomg trinh x^ - V3x^ + 7x - N/3 = 0, do ta phan tich A : ' - 3V3x'+ 7x - ^/3 = - V x ' - 2V3x'+ 2.3x + X - V3 = "x = V3 <=>(x-V3)(x'-2V3x + l ) = x-S =0 x = V3+V2 x'-2V3x + l - x = V3-V2 5) Khai trien ve trai roi nit gon ta thu dugc phuang trinh 2x^ + lOx = 12 <^ + 5x - = « - < » ( x - l ) ( x ^ + j c + 6) = 0<» + - X + 6x - = "x-l =0 x^ + X + = o x = l Vi du Tim m dk phuang trinh x^ - (3 + 2m)x^ + 5mx + 2m^ = c6 nghiem phan biet Lai giai Ta c6 x^ - (3 + 2w)x^ + 5/nx + 2w^ = X = <=>(x-2m)(x^-3x-/n) = 2m x ' - x - m = 0(2) De phuang trinh c6 nghiem phan biet va chi (2) c6 hai nghiem phan bi?tkhac 2m <:> {2mf m^O;m^ — -3.2m-m¥=0 uiifi,^ -(Ism -9 A = + 4/7j>0 Vi du Giai phuang trinh (x^ + 3x - 4)^ + 3(x^ + 3x - 4) = x + L&igidL Ta c6 (x^ + 3x-4)^ + 3(x^ + x - ) = x + 38 ( x - l ) ( x + 4)] +3(x-\)(x + 4) = x + o ( x + ) ( ( x - l ) ' ( x + 4) + ( x - l ) - l ) = (x + 4)(;c' + 2x' - 4x) = <=> X ( x + 4)(x' + 2x - 4) = o x = 0;x = -4;x = - l ± V , ^ , ^^ ., p Vay nghiem ciia phuong trinh la x-0;x 4;x \±^ Vi du Giai hk phuong trinh (x +1)(x + 2)(x + 3)(x + 6) > 35x' L&igidL Taco (x + l)(x + 2)(x + 3)(x + 6)>35x' (x + l)(x + 6)][(x + 2)(x + 3) >35x' « ( x ' + 7x + 6)(x'+ 5x + 6) > 35x' o ( x ' + x + + x)(x'+6x + 6-x)>35x^ o ( x ' + x + ) ' - x ' >35x' « ( x ' + x + 6)' >36x' o ( x ^ +6x + 6-6x)(x^ +6x + + 6x)>0 <::>X^+12x + 6>0<::> (x^ +6)(x^ +12x + 6)>0 "x<-6-V30 x > - + V30 Vay nghiem cua bdt phuong trinh ( - 0 ; - - V30 j U (-6 + V30; + QO j Vi dy Giai bdt phuong trinh (4x + l)(l2x - l)(3x + 2)(x +1) < lo^gifli Taco (4x + l ) ( l x - l ) ( x + 2)(x + l ) < , (4x + l)(3x + 2)JL(l2x - l)(x + 1)J < o (l2x' +1 Ix + 2)(l2x' +1 Ix -1) < 12x^4-llx + - + 2 x ' + l l x + - - - <4 2 \ 12x'+llx + — <4 <=> 12x^+llx + - - ^ < 4 2J • - .y 12x'+llx + - - 2 \2x'+\lx + 2- +2 <0 Ix - 2)(l2x'+ Ix + 3) < (12JC'+ Vi 12x^ +1 Ix + > 0, Vx e M nen hk phuomg trinh tuomg duang vai io ^ ^ - I I - V T<7 x < I I + V2T7 12x +llx-2<0<=> 24 24 Vay nghiem cua bit phuang trinh la ^-11-721? -11 + V217^ 24 24 Vi du Giai phuang trinh (x - ) ' (l + 2x + 3x' + + 2015x'°") = L&igidi Dat = + 2x + 3x' + + 2015x'''"' (1) Phuang trinh da cho tra (x -1 )^ = Taco x5 = x + 2x^ + 3x^ + + 2015x^'"\ Suy (x - ) = 2015x''"' - (l + x + x' + + x'°''), hay Do (x -\fs^{xl)(2015x^°'' _ (1 + ;c + x^ + + x'°'^)) (2) Tir(l)va(2)tadugc ( X - 1) (2015X^°'^ - (l + ;c + X ^ + + x^""")) = 2015x'''''- x^°'^-0 = «(x-l) x-1 o(x-l)2015x^'"^-x^°''+l = l x=0 ox^'"'(2015x-2016) = 0<» 2016X = 2015 Bai tap phan 2.1 Bai Tim gia tri ciia tham s6 m dk phuang trinh x" -2mx^ -x + m^ -m = CO b6n nghiem phan biet Bai Tim diSu kien cua a dk hai phuang trinh ' ' , x^ + 2x + a = va x^ - 4x - 6a = CO nghiem xen ke 40 Bai Giai bat phuang trinh - - Ox^ - Ox + > ( l ) Bai 4, Giai phuang trinh vai m la tham so thuc : (x+ \) = 2(x + m){x + Im) Hu-ong dan bai tap phan 2.1 Phuang trinh da cho tuomg duang voi x^+x + l - / n = (1) (x^ + X + - - X - m) = <: x'-x-/w = (2) Phuang trinh da cho c6 bon nghiem phan biet va chi hai phuang trinh (1) va (2), moi phuang trinh c6 hai nghiem phan biet va hai phuang trinh khong CO nghiem chung , Dieu kien de hai phuang trinh c6 hai nghiem phan biet la [A, = / « - > /n > — Aj = 4m + > Gia su hai phuang trinh (1) va (2) c6 nghiem chung la x^ thi x„ + x„ +1 - /« = =>2Xn+l = < = > X n = =>/« <=> " XQ - X O - W = = - Tir de phuang trinh da cho c6 bon nghiem phan biet thi m> — Hai phuang trinh da cho c6 hai nghiem phan biet va chi A' = l - a > A' = + 6a > Goi x,,X2 (x, < Xj) la hai nghiem cua phuang trinh thu nhat va X3,X4 (xj < X ) la hai nghiem cua phuang trinh thu hai Dieu kien de mot hai nghiem X3,X4 nkm (x^Xj) va nghiem namngoai (xpXj) la ( x j - x , ) ( x - X ) ( x - x , ) ( x - X j ) < Theo dinh ly Viet ta c6 x, + X2 = -2,XiX2 = a,X3 + X4 = 45X3X4 = -6a 41 Dodo {x^-x^)(x^-X2) = Xj-(x^+X2)x^+x^X2^xl+2x^ (x^-Xi){x^-X2)^xl-{x^+x^)x^+x^X2=xl+2x^ Do {x, -X,)(x3 <=> (xj + -x^){x, + + +a +a ^ ^ i v i Ufi "'-^ -x,){x,-x^)<0 + a) < <=> (x3X4)^ + 2X3X4 (X3 + X4) + 4X3X4 +a(^xl + X4) + 2a(x3 + X4) + : <0 <^ 49a^ - 48a < O o O < a < ^ (thoa man diku kien) Vay < a < 48 49 Ta CO x ' - x ' - l O x ' - Ox + > o ( x ' - 7x + 2)(x^ + 2x + 2) > <:>x-7x + > o x < 7-V4T Vay nghiem cua b i t phuong trinh la x e -vx > -co; + V4T 7-V4I u 7+V41 -;+oo Phuong trinh da cho tuong duong voi x'+x'-2x'-6mx-4m'=0 a < » w ^ + x m - ( x ' ' + x ^ - x ^ ) = Xem day la phuong trinh bac doi voi an m va x la tham s6 ta c6 A ' = x ' + ( x ' + x ' - x ' ) = x ' ( x +1)' > x' x' X T u m, = x; m, = ' ' 2 Do phuong trinh da cho tuong duong voi „2 \r m +—+X x' x^ + — = m Tirdo xM.2 , , = - l ± V l - m ; X34 = - ± - V l + 8m voi m e '3,4 2 8'2 42 V o i m<- phuang trinh c6 nghiem = - ± yjl-2m ' ,.5,, / j A i -' -f V o i m>— phuang trinh c6 nghiem x^^ - — ± — V l + 8/n ,2 • 2 2.2 Phuang phdp dat an phu dai so V i du Giai phuang trinh + (Di thi tuyin sinh THPT Chuyen Dai hoc Vinh 2014) XTtO Lm giau Dieu Icien j ^ ^ <i> <{ x * l Dat / = x^ - 2x K h i phuang trinh da cho t r a r=l t t+i ^ 3- ^ : V a i t = l, t a c o x ' - x - l = c ^ x = l +V2 Vai ^ = taco x ' - x + = < ^ x = V i du Giai phucmg trinh ( x +1)^ L&igidt*) Bihukien: x+- (t/m) (t/m) ^ ^ = (x^ + ?) x^O ' Chia ca hai ve cua phuang trinh da cho cho x^ ta c6 2+ ^ Xy 2Vr 2+ - Xj x+— x+ - V 3^ x+ — = Xy X f V X Xy X + - + - Dat a = - ; = x + - T a c (2 + af X X \ b = (b + 2af <^a'b + 'Xb^b'+Aa\ <^a\b-l) = 4b{b-\)<^ = a'=b Vaife = 4<=>x + - = 4<z>x^-4x + = 0<» x = \ x = 3' Vai X =b \ai x + - = — <:^x^ + x - = 0<»(x-l)(x^ + x + 4) = 0<::>x = l Vay x = l;x = Vi du Giai phuong trinh (x' - 4x + 5)^ - (x - l)(x - 3) = L&i giai Ta c6 (x' - 4x + s)^ - (x - l)(x - 3) = <:^(x'-4x + ) ' - ( x ' - x + 3) = c>(x^-4x + 5)^-(x^-4x + 5) = *' \t = -\ Dat/ = x^-4x + 5,tac6 ^ ^ - f - = 0<^ [r = Vai t = -\a CO x^ - 4x + = -1 <::> x^ - 4x + = (v6 nghiem) Voi t = taco x^-4x + = < ^ x ^ - x + = 0« x = l x = 3' Vay nghiem cua phuong trinh la x = 1; x = Vi du Giai phuong trinh (x - 2)^ + (x - 3)" = LM giai Dat / = x - ta c6 r=0 ; = -l Vay nghiem cua phuong trinh la x = 2; x = ' Vi du Giai phuong trinh 2(2x^ - 2x - 5)^ - 4x' + 3x + = v2 Lai giai Dat j = 2x' - 2x - 5, j = \ X — V 2j X, si x=3 x = 2' -il>_il 2 44 Taco -4x^ +3x + = -2y-x-5 Phuong trinh da cho trd 2y^ -2y-5 = x ^2x^-2x-5 = y Tu ta CO he phuong trinh [2y'-2y-5^x Suyra {x-y){2x + 2y-l) = 0<:> x = y x = -l 5• X = — taco 4x^ - 2x-\\ 0c^x = 111^^ Voi = X ta CO 2x^ - 3x - = <=> Yai y = x Vay nghiem ciia phuong trinh la x = -l;x = —;x = ^ - - ^ ^ Vi du Giai phuong trinh 8(x^ + 2xf - x - = LdigidL Ta c6 8(x^ + 2x)' - x - = 0<r>(2x^+ 4x)' = x + Dat ay + = 2x^ + 4x Taco ^^{ay + bf c^2a^y^ +4aby-x + 2b^-3 = 2x^ +4x-ay-fe = Ta CO he a V + a ; ; - x + ' - = Chon a = = 1, ta CO he 2x^+4x-y-l = 2;;^ + 4:i; - X -1 = 0" Trir ve theo v8 ta c6 (x - y){2x + 2y + 5) = Voi y = X ta CO 2x^ + 3x -1 = X = 45 ^ CO 2x^ ^ +4xA + 2x + 1, =^0 Vcfi y = 2x +—5 ta <=>4x^+10x + = 0<»x = -5±Vl3 u ^ U A da~ cho u co' nghiem u ^ x = -3±Vl7 va, x = -5±Vl3 Vay, phuong trinh 4 Vi du Giai phuong trinh - + - x + = L&igidL Ta CO x' - 6A:' + 15A:' - 18x + = <:> [x' - 3xf + 6(x' - 3x) + = Dat / = x' - J C , ta CO f' + 6? + = <:i> t = - l / = -5 Voi / = -1 <=> x' - 3A: + = <=> x =3±V5 Voi / = - o x ' - x + = (VN) Vay nghiem cua phuong trinh la x = 3±V5 Vi du Giai phuong trinh (x' - x +1)^ - 6x' (x' - x +1)^ + 5x^ = Ldi giai Voi x = 0, khong phai la nghiem cua phuong trinh Voi x^Q, chia ca hai ve cua phuong trinh cho x'^ ta c6 _ ^ (x'-x + l)'-6x'(x'-x + l)'+5x'=0 ^^-X+1 X Dat / = ' X - X +1 Ta CO - ^ r - x' - X + + = 6r + = <^ ?=1 /=5 x ' - x +l = x'-x +l_ ^ x ' - x + l^' Voi ? = x'-2x + l = o x =l x ' + l = 0(VN) 46 Vai / - <:> ' X - X -x +1 +1 x'-x +l = -4l x'-(>/5+l)x + l = 0<:^X = ^ x'+(V5-l)x + l-0(VN) Vay phuong trinh c6 nghiem jc = 1, x = V5+l±j2(V5+l) Vi du Giai phuong trinh x = - 2014(l - 2014x^)\ x = 1-2014/ LM giai Dat jj; = - 2014x^ Ta c6 he ;; = l-2014x'' Tru ve theo vk ta c6 [x-y)^ 2014(x - >;)(x + <::> (x - >;)(l - 2014(x + y)) = Voi jj^-x taco x = l-2014x^ <=>2014x^ + x - l = -1 + V8057 /ft., <=>x = ^, (thoaman) 4028 Voi l-2014(x + >;) = =0<»0 < zv>=x — x taco ,— x = l-2014x' «2014x'-x-—— - 1±A/8057 (thoa man) 2014 4028 ^ ^ 2014 2014 , bat phuong , , , ,la, x,= -l±V8057 Vay nghiem cua trinh — , x = l±>/8057 4028 4028 Vi du 10 {Tdng qudt Vi du 9) Giai phuong trinh x = - a (l - ax^)2 , a la hSng s6 Lofi giai Dat y = - Ta c6 he X=1 - ay^ y = l-ax' 47 Day la he doi xung loai 2, each giai thuong tru ve theo ve ta luon c6 nhan tu chung x-y Bai tap phan 2.2 Bai Giai phuang trinh x +3^2-3x^1 = 'Bai Giai phuang trinh (^4x^ - llx + sf ^2x + \ Bai Giai phuang trinh + 9)(x + 9) = 22(x - ) \ x^ + x Bai Giai phuang trinh ^ = ( x ' - x + l) Bai Giai phuang trinh lx{lx^ + X + 3) +13x(2x' - 5x + 3) = 6(2x' - 5x + 3)(2x' + x + 3) Bai Giai phuang trinh x^(x-l)^ =(2x-l)^ + I Bai Giai phuang trinh 2(x - 3)^ (x + 2)^ = (2x -1)^ - Bai Giai bat phuang trinh x^ + 3x^ + 2xV 3x +1 < (l) Bai Giai phuang trinh (x-2)(x^ + x - ll)^ = (Sx^ - lOx + ) \ {m tuyin sink THPT Chuyen Lam San Thanh Hoa 2013) Bai 10 Giai phuang trinh (x^ + l)(5x^ + X + 8) = 22 (x + 3)^ Bai 11 Giai phuang trinh x"* + 6x^ + 5x^ - 12x + = Bai 12 Giai phuang trinh x" - 4x^ - 2x^ + 12x + = Hirang dan bai tap phan 2.2 il.Dat >; = - x \ T a c h e jx ++ 33x'/ ==22 ( ) ' Trir ve theo v6 ta eo (x-y)[\-3x-3y) = 0<:> y = xl-3x y=— 48 / 3 X x=- l Vdi y = x thay vao (2) taco jc + 3x^ = « > J C ^ + x - = o x=— = ^ - JC thay vao (2) ta c6 Voi - - x +3x^=2«9x^-3x-5 =0 « x = i ^ ^ Vay nghiem ciia phuomg trinh l a x = - l ; x = - ; x = 2.Dat 4x^-12x + - q y + Z).Tac6 + Lai =2x + l<i>aV+2a6>'-2x + 6^-l = - 12x - ay + - = CO Ta C O h? aV+2a6>;-2x + ' - l = 4x^-\2x-ay + 5-b = Chpn a = 2; = - , ta CO he 4/-12>'-2JC + = [ X ^ - J C - ; ; + = 0' Trir vg theo vktaco {x-y)[4{x Vdi >; = x taco + y)-l0) = 0l<=> 2y = 5-2x' 7±y/l7 -14JC + = 0<=>2x^-7x + = 0<=>x = Voi 2>' = - x taco 4x^-\2x-{5-2x) + ^ = 0^x Vay nghiem ciia phuang trinh la x = '^-^ Ta C O x{x' +9)(x + 9) = 22{x - if ^(x' y=x +9)(x' +9x) = mt a = x^ +9x,b^x^ +9.Tac6 22{x-lf a-b x-l = = ^x= 5±Vl3 Do phuong trinh da cho tro \ ab = 22 a-b «22fl^-125a6 + 22r = ) Ta CO > 0, chia ca hai v l ciia phuong trinh cho b^ nen ta c6 a_2^ a b~\\ 22 -125 + 22 = 0<» a l l lb~ Vai - = —<:^lla = o l l ( j c ' + x ) = 2(x'+9) b 11 <=>9x^+99x-18 = O X ^ + 1 A : - = 0<::>X = Voi | = y < : > ( x ' + x ) = ll(x'+9)<»9x'-18x + 99 = (v6nghiem) -\\±yf\29 Vay nghiem cua phuong trinh la x = Voi X = 0, khong phai la nghiem ciia phuong trinh Voi X 7t 0, chia ca hai vS ciia phuong trinh cho x^ ta c6 x +- ^ ^ = 2<=>x + - = \2 X-1 + X-1 + Bat t = x + l,tac6 t + \^2t^ ^2t^ -t-\ 0<^ X ;:i-:<^'itwr ^~~2 ^ SJ-^ijiJ F Vai r = <=>x + - - l = - - o x + - = -^ (vonghiem) V6ir = l o x + - - l = l o x + i = 2<:>x^-2x + l = 0<=>x = l X X Vay nghiem cua phuong trinh la x = ft i - 50 R6 rang x-\; x-—; x = khong phai la nghiem cua phuomg trinh Vai xit\;x^-;x*0 Chia ca hai ve cua phuong trinh cho {ix^ -5x + 3^(2x^ + x + 3), 2x \3x = ,6<=>2 13 + —z + taco = 2x-5 + - 2x + \ ~ 2x^-5x + 2x^+x + X X t =\ D a t / - x + - T a c ^ + ^ = o ? ' - / + 33 = 0« 11 X t-5 t+l t= Vori t = l taco 2x + -^\<^2x^-x X Vai ^ = — ta CO 2x + - = — o X + = (v6nghiem) - Ix + = 'fi x=2 3X= — i') *'1 Vay nghiem cua phuong trinh la x = 2; x = — =(x + x - l ) % Taco x ^ ( x - l f = ( x - l ) V <»x^(x-l)^ Dat a = x;Z) = x - l T a c a - = l Dodo taco (abf ={a + bf +2 = (a-bf +4ab + = 4ab + 'ab = 2-yll Ta CO {ab)^ -A(ab)-3 = <=> ab^2 + yll ^^rS ' Voi a6 = - V , t a c x ( x - l ) = - V < : ^ x ' - x - ( - V ) = 0(v6nghi?m) Vai a6 = + 7?,taco x ( x - l ) = + V ^ o x ^ - x - ( + V^) = 0<=^x = i^^/^±^ Vay nghiem cua phuong trinh la x = 1±V9 + 4V7 l.Bat a = x-3; b = x + 2.Tac6 \ „ / 2a^b^=(a + b) -9^{a-b) ^{abf Voi ab = -2ta -2ab-S a + b = 2x-l, , x2 +4ab-9 a-b = -5.T\jdo = 25 + 4ab-9 = 4ab + \6 = 0<:> ^^^^ \_ab - -2 \a=b-5 CO he a^b-5 < <^<t, • [ab = -2 b= Tirdo b^-5b + = ^ [{b-5)b 'a = b-5 <=> i = -2 s-Vn X = + Vi7 b-5 +2 =0 b-5 a = 1(6-5)6 =4 ab^4 5-V41 Y* — A — ^ + 741 V — [b 1-V41 + ^41 X — , + Vl7 X- a = \b'-5b 2 0o b= a^b-5 Voi afe = ta c6 he - b= Tirdo ' - - = o b= i+VT? Vay x = + V41 , x= 2 Voi jc = 0, (l) <=> < khong la nghiem cua bdt phuong trinh, Voi X ^ 0, ta CO > 0, chia ca hai v6 cua (1) cho x^ ta c6 (1)<»JC'+3JC + + - + ^<0 Datr = x + - , r > , t a c / ^ + / < < i > - < r < < » - < x + - < X x^+1 X x+ -<0 <0 [x<0 X x'+3x + l x+ ->-3 X |x'+3x + l < >0 52 ^ -3-V5^ ^ -3-V5 Vay nghiem cua bit phuang trinh la ——— <x < ^'^^ Dat y = x-2, phuang trinh da cho tra y[(y + 2)' + 6( J + 2) -11J ^y(y^+\0y ^y' =[5(y+ + 5f =[5y^+\0y ~* ' ' 2)'-I0{y+ 2) + l + \f is'J -5y' +I0y^ -\0y^ +5y-\ <=>{y-\f =0 y = l<^x-2^\<:>x = Vay nghiem cua phuang trinh la x = 10 Taco ( X ' + ) ( J C ' + J C + 8)=:22(X + 3)' <»(x' +l)(5(x^ + l ) + (x + 3)) = 22(x + 3)' o 5(x' +1)^ + (x^ + l)(x + 3) = 22(x + 3)^ Vai jc = -3 khong phai la nghiem Voi x^-3 chia ca hai ve cua phuang trinh cho (x + 3)^ ta dugc +P x+3 + X x^+r - 2 = x^+1 Dat r = T a c o r + r - 2 = 0<» x+3 x+3^ tt == 2—11 • x^+1 Vai ? = <^ x + = o x ^ - x - = 0<»x = l±%/6 Vai / = < ^ ^x^+- 3^ = -—<=>5x^ + l l x + 38 = \(v6nghiem) ) Vay nghiem cua phuang trinh la x = ± Il.Taco x'+6x^+5x'-12x + = ^(x^+3xf-4[x^+3x) +3=0 D$t/ = x^+3jc,tac6 ? ' - f + = 0<» t = \ t = 3' Vai t = \a CO Voi ^ = taco A/' ' + 3x = <=> + 3x - = X= -3±ylu +3x = 3<=>x^ + x - = < ^ x = U ' V , n - + 713 -3 -3 Vay nghiem cua phuong trmh la x = Ta CO x' - 4x^ - 2x^ + 12x + = + V2T + V2I ,x = , 12 v <»x'-4x^+4x'-6(x'-2x) + = <»(x'-2x)'-6(x'-2x) + - [t = Dat/ = x ^ - x , t a c - / + = 0<^ Lr = Vai ^ = taco x ^ - x = 2<=>x^-2x-2 = 0<=>x = l + V3 Vai r = taco x ^ - x = 4<=>x^-2x-4 = 0<=>x = l±V5 Vay nghiem cua phuong trinh l a x = : l ± V , x = l±V5 -i' 2.3 Phuong phdp luang gidc hoa Vi du Giai phuong trinh 8x^ - 6x - = L&i giai Nhan xet: Day la phuong trinh bac ma nghiem khong thl nhim dugc bang may tinh mot each thong thuong Ta biln d6i nhu sau x ' - x - l = 0<»4x'-3x = - Ta CO ^ = cos y = cos^ ^ - cos ^ Do phuong trinh tuong duomg vai 4x^ - x = 4cos^ —-3cos— 9 n 4cos^ — - = 4x^ +4xcos—+ 9^ x-cos54 Xet phuong trinh 4x +4xcos—+ 4cos = c6 A' = 12sin nen phuong trinh c6 hai nghiem x- —1 cos — ±—sin — = cos(n— +.In • 9 ) Tu phuong trinh c6 cac nghiem x = c o s ^ ^ s v a x = cos — ^ Vi du Giai phuong trinh x ( x ' - l ) ( x ' - l ) ' = - - (l) L&igidL Di6ukien x^O i Nhdn xet Ta c6 cong thuc cos^x - = - sin^ x; cos 2x = cos^ x - , bilu thuc ve trai c6 su xuat hien ciia x^ - va 2x^ - nen ta c6 suy nghl c6 the don gian bai toan hon bang each dat x = c o s a , muon vay truac het ta so sanh x voicacgiatri -1 va Phuong trinh da cho tuomg duomg voi x ^ ( x ^ - l ) ( x ^ - l ) ' = x - l (2) J Voi X = la nghiem ciia phuong trinh Voi X ^ (2) tuong duong voi 32x' (x + l)(2x' - ) ' = (3) Voi X > thi 32x^ (x + l)(2x^ -1)^ > nen (3) v6 nghiem Voi x < - l thi 32x'(x + l ) ( x ' - l ) ' < nen (3) v6 nghiem Voi -1 < X < Ta quay lai phuong trinh (1) Voi x e ( - l ; l ) , dat x^cosor, a e ( ; n) Ta c6 phuong trinh da cho tuong duomg voi 32cosa(cos^cir-l)(2cos^a-l)^ =1 COS or <=> 32 cos^ a sin^ a cos^ 2a = - cos a <» sin^ a = - cos or kin Sa - a + kin- <:> cosSor = cosa <=> 8a = - a + k2n kin ,k^Z Vai a = ^ Voi « = ^ v i « G ( ; TT) nen A: = 1,2,3 Do a = ^ ; ^ ; ^ v i as(0;;r) nen it = 1,2,3,4 Do a = ^ ; ^ ' , ^ ; ^ 9 9 Kdt hgrp lai cac nghiem ciia b i t phuong trinh la f, ITT ATT 6n In An dn x e | l ; c o s — ; co5 —;cos—;cos—;cos —;cos—;cos— V i dy Giai phuomg trinh - 3= 3x^-1 V3x L&i gi&L DiSu kien x^^O Nhan thiy x = ± - ^ khong phai la nghiem cua V3 phuomg trinh Ta c6 x^ - = Dat X 3x^-1 V3x x'-3x 3x'-l V3' = tan « , a e Tucongthuc tan 3a = taco tsai3a = -j=^3a >/3 Do a e 2'2 tan a r - t a n a 3tan^a-l = — + kn:=>a = — + — 18 ,ksZ n ^ n kn ^7t ^ , ^4 , < — + — < — <::> — < A: < - => ^ =-1,0,1 18 3 nen « = - — ; — ; — Do phuomg trinh c6 cac nghiem 18 18 18 Sn ^ n In x = tan ;x = t a n — ; x = t a n — 18 18 V i du T i m tat ca cac so thuc 18 x,y thoa man - ( x V + / z + z^x) + 6(xv + j z + zx) = 56 L&igidL Phuong trinh da cho tircmg duong voi -3x = y _ 3;c - yf + (y' -3y- z)' + (z' -3z-xf =0<^ /-3y z'-3z NSu x>2 thitu y = x^-3x = x(x^-3)>2.Tuomgtir = z =x z>2 Cpng ve theo ve ba phuong trinh cua he ta dugc x'-4x + y'-4y v6 ly vi + z'-4z = x[x^-2) + y[y^-2) + z[z^-2) = 0, x,y,z>2 N6u x<-2, tuomg tu ta c6 y,z< -2 va din d€n v6 ly Voi - < x < , dat x = 2cosa,a s[0;n] Khi y = ( cos^ a - cos a ) = cos 3a • z = 2(4cos^3a-3cos3a) = 2cos9cir X = ( cos' 9Qr - cos 9a) = cos 27a \21n = n + k2n Do cos 27a = cos a <=> " <=> 21a^-a + k2n: 13 a = 14 Voi a = — , vi a G fO;;r nen 13 ^ ae>^ ' ' ' ' ' ' ' ' ' ' 13 ' 13 ' 13 ' 13 ' ^ kn Vai a = — , vi a e fO;;;rl nen 14 ^ •' - n 2n 3n An 5n In Zn 9n \Qn Un 14 14 14 14 14 14 14 14 14 14 V \2n 14 \3n 14 J Do CO 27 bo nghiem can tim la (2 cos a, cos 3a, cos 9a ) vai a nhan 27 gia tri hai trucmg hap tren 57 Vi du Giai phuong trinh Sx(2x^ - l)(8x' -8x' +1) = Ldi giai Tir cac cong thuc cos 2A: = cos^ x - , cos 4x = cos" x - cos^ x +1 xult hien ta nghi toi viec luong giac hoa phuong trinh bang each dat X = cos or, muon vay truac het ta so sanh x voi cac gia tri -1 va Voi X > thi 8x(2x^ - l)(8x'' - 8x^ +1) > nen phuong trinh v6 nghiem Voi X < -1 thi 8x(2x^ - l)(8x'' - 8x^ +1) < nen phuong trinh v6 nghiem Voi -1 < X < Dat X - cosa, a e (O; Khi ta c6 phuong trinh tro cos a cos la cos Aa -1 •o sin a cos a cos la cos Aa - sin a (vi sin or ^ 0) kin <=>sin8Qr = s i n a o " _ <=> ^ _ ,k&'L ^a-n-a + kin: n + kln a^2;r , / ' A \ I '1 -5 _ In An 6n ^ V 1 \ , «,^„ n 3n 5n In n + kln , Vai a = VI aeiO; n) nen A: = 0,1,2,3 =— ;—;—;— ^^ 9 9 Tu phuong trinh da cho c6 nghiem la In An 6n n \ In} cos — ;cos —1 ;cos—;cos—; —;cos—;cos— 9 J> Vi du Giai phuong trinh 16x' - 20x' +5x = n, voi n nguyen duong Ldi giai Chung ta xet hai truong hop sau: |A:|<1 Dat x = cosa, a €[0;;7r], phuong trinh da cho tro cos5a = n (1) +) Voi « > 1, khong thoa man 5 f In An 1( ,t ^Q,t ^+\ Khi phuong trinh da cho tro +) X > Dat X = - t + Vai n = 1, (1)21CO nghiem x - cosa,a ^ \t 0' f 1^ ' + 5- ( —f Kt + -t) t + - { t + -t) - n \ T - <=>t'°-2nt'+1^0 <=>t' ^n±^n^-I Suy x = — Tom lai, vai « = phuang trinh da cho c6 nghiem X = cosa,a s<0 ' ; vai n > phuang trinh da cho c6 nghiem , X nhat X- — Bai tap phan 2.3 Bai Giai phuang trinh 4x^ -3x-m vai m la tham so va |w| < Bai Chung minh rSng vai cac gia tri ciia x thoa man - l < x < l thi bit phuang trinh sau luon dung vai cac so nguyen duang n {i+x)"+(\-xy<r Bai Tim t4t ca cac s6 thuc x,y,z thoa man x y z y z X —+ — +— + - +- + - + x'+y'+z'-4 yxy yz zx• / 1 = Bai Giai phuang trinh 2(4x' - 3x)(4x' -1) = ( l ) Huong dan bai tap phan 2.3 Giai phuang trinh 4x^ -3x = m vai w la tham s6 va |m| < Vi w <lnentadat/n = cosQr,khid6tac6 7w = cosa = 4cos3 ay - 3^ c o say Phuang trinh da cho tuomg ducmg vdi 4x - x = 4cos x-cos- 3cos— a 4cos^ —-3 - 4x^ +4xcos—+ Xet phuomg trinh +4xcosy+ c o s ^ ^ - = c6 A' = 12sin^^ nen phucmg trinh co hai nghiem x- — cos — ±—sin — = cos 3 Vay phuong trinh c6 ba nghiem la x = cosy, x - cos ^ Chu y: Doi voi bai ta c6 the dat x = cos a r6i tir ta c6 cos 3a = m Tuy nhien each giai bat buoc ta phai chung minh x| < va phSn lay cong thuc nghiem rac roi hon Chung minh rSng voi cac gia tri ciia x thoa man -1 < x < thi bdt phuong trinh sau luon dung vai cac s6 nguyen duong n (l + x)"+(l-x)"<2" Dat X - cosa, a E [O; TT] Ta c6 s i n2' a(l + x)" + (l - x)" = (l + cosar)" + (l - cosa)" = cos a— 2J = 2" cos — + Sin — <2" ^ a « ^ = 2" 2) COS —+ sin — Dau dang thuc xay x = -1 hoac x = Ta c6 dieu phai chung minh Tim tat ca cac s6 thuc x,y,z thoa man p = x ^ + y + z ^ - — + — + — +4^ X y z + 5'I « M \xy yz1 Pzxj • « - +- +- y Phuong trinh da cho tuong duong voi X y z + r —1 + —1 + — = x ' + / + z ' - f + 1— + 2—X ; + yz {y- + ^- + -xj _J _2 X r 2^ 2^2 2^ X - - + — + —+ + — + — = 0« V zI X yj V y- y z X) y _J _2 y 2' _j _2 z X 60 Taco x = ±l,>' = ±l,z = ±l khong phai la nghiem nen he tuomg duomg voi Ix \-x' z X- Dat X _2:y_ • \-y \-z' va tan or, tan2Qr,tan4a ?t = tan a vai or' Ix V a i x = tanQr thi y- tm2a Tirdo z = tan4a va x ^ t a n S o r D o d o t a c o tanSa = tana<=>or = — , A : e Z Vi as 2, nen - — < — < —•c:>-3,5<«:<3,5 • Are{-3,-2,-1,0,1,2,3: ^ [ Dodo as\ Zn In n ^ n In ;—;0;—;—;— In Tir phuong trinh da cho c6 nghiem {x,y,z) = (tan or, tan 2cir, tan a ) vai a nhan cac gia tri tren Giai phuong trinh ( x ^ - x ) ( x ^ - l ) = = l ( l ) Vai |x|>l, ta c6 2(4x^-3x)(4x'-l) 2x 4x^-3 ^^^^ 4x'-l > , nen (1) v6 nghiem Vai - < x < l Dat x = c o s a , a e (0;;r) Tac6 (4cos^a - 3cosa) (4cos^a - ) = (4cos^Qr - 3cosa) ( - sin^ a ) = Vi s i n a ^ O nen taco ( c o s ' Q r - c o s a ) ( s i n a - s i n ^ a ) = sincir 61 <:> sin6cir = sina kin - '~J> :fi a5 7t klTC a = —+ 7 Voi a = ^ , d o a e ( ; ; r ) nen < — < ; r = > < A : < - n e n A: = l , 5 , - - ' In An Suy X = c o s — : x = cos — ^ 5 ,, - ^ k2n , „, Vol a = y + — , t u o n g t \ r , k = 0,\,2n 3n 5n Suy X = cos—; x - cos—-;x = c o s — ^ 1 Vay phuong trinh da cho CO nghiem fi ir-rxi V 2;r 4;r n x = c o s — ; x = c o s — ; x = cos — ; 2.4 Phuomg phdp ddnh gid 3n X = COS—;x 5n = cos— nt - ' - ^ J« - Vi du Giai phuong trinh x^ + 3x + = Ldi gidL Ta kho hy vong c6 thk tim thSy duoc nghiem cua phuong trinh voi cac bucfc tinh toan thong thuomg BiSn d6i ta c6: X^+3x + l = 0<::>X^+l = -3x Chpn a,b la hai s6 cho: a^+b^ = 1; ab = - l Ta c6 a^+b^ +x^ = 3abx Xet h3ng dang thuc a' +b' +x' -3abx = -{a + b + x)((a-bf Dodo +b^ +x^ ^3abx<^ +{b-xf +(x-af) a+b+x=0 a=b= x Ta CO a^ b^ la nghiem cua phuong trinh -X-\ 0^a^,b^ = 62 Lay a = i j ^ ; b - n-—(a^b) Khi phuong trinh da cho c6 nghiem duynhatia x = -{a + b) = - \ Vi du Giai phuong trinh {x + if + (x + yfs)' = 18 -8N/5 , ^ ,a'+b\fa Lai giau Ta co > V (x + i f +[x + VS)' +b ^ - + N/5^ voi a + > , « e N , « > / (-X Dodo Taco \ -i+Vs - i f + (x + >/5)' ^ f - i + V I \ >(2-^/5)'=9-4^/5 Do (x + i f + (x + A / S ) ' > 18-8^5 Ddu • = ' xay va chi - x - l = x + >/5«>x = • V?iy nghi?m cua phuong trinh la x = Vi du Giai phuong trinh (Sx - 4x' - l ) ( x ' + 2x +1) = 4(x' + x +1) L&i giai Ta c6 voi x = - khong la nghiem ciia phuong trinh Vori X 5t - , ta CO phuong trinh tuong duong voi 8x-4x - Dat>' = x^+x + x^+2x + r Khido < » x ^ + x + l = v(x^+2x + l ) x'+2x + l ^ X^ + X + < ^ ( l - ; ) x ^ + ( l - > ' ) x + l - y = Voi y ^ l phaico A > o ( l - > ' f - ( l - > ' f >0<:>4>;-3>0<=>>'>- ^ x^+x + ^ Do >- x^+2x + l 63 ^ , 8x-4x - ^ , ,v2 3 4 ^ 4 +1 Tu de phuorng trinh c6 nghiem x ' + x + l <::>X = 8x-4x'-l Vi du Giai phuomg trinh (^^ +1) x ' + x - l + x ( l - x ^ ) = ( x ^ + l ) ' ( l ) LcdgiaL Xet hai truong hop a)x^+2x-l>0op ^~f _ x > - l + V2 Ta CO (1) tro [x^ + l)(x' + 2x -1) + 6x(l - x') = (x' +1)' <»2x'+x'-4x + l = c ^ ( x - l ) ( x ' + x - l ) = 0<:> x = Thu lai X = thoa man b ) x ' + x - l < < ^ - l + V2 < X < - + N/2 Ta CO (1) tro - ( x ' + l)(x' + 2x -1) + 6x(l - x') = (x' +1)' o x * + x ' + x ' - x = <»2x(x + l)(x' + X - ) = 0<» -3±Vl7 • X = v x = -1 -3±Vl7 • X= Thu lai X = 0;x = - thoa man Vay nghiem ciia phuomg trinh la x = - ; x = 0; x = r p3 Vi diji Giai phuomg trinh 16x(x^ + 3) = 3x + LM giai Dieu kien x Ta c6 16x(x'+3) = 3x + - 16x^(x^+3) = ( x ^ + l ) \ 64 Ap dung b i t ding thurc vai a,b,c khong am abc < \6x'{x' +3) = {4x'){4x'){x' +3)< De dau bang xay phai c6 4x^ = fa + b + c^' Ux'+4x'+x'+3^' +3 ta = CO {3x^.l)\ jc^ = «> x = +1 Vay nghiem cua phuang trinh la x = ± Bai tap phan 2.4 Bai Giai phuang trinh 256x' ( l - 3x) = Bai Giai phuang trinh ' x ' - x ' - x + 40^' Bai Giai phuang trinh ^2x^-llx + 2l'' = 4x + = 4x-4 Bai Giai phuang trinh (x^ + x -1)^ - x^ + 3x^ - = Huang dan bai tap phan 2.4 Ap dung b i t ding thurc abed < a + b + c + d^'^ x ' ( l - x ) = x.x.x(l-3x)< ta CO x + x + x + l-3x 256 1 Dau bang xay va chi x = — Vay nghiem ciia phuang trinh la x = — ' ' Tu gia thiet ta c6 x > - Ta c6 ^ x ' - x ' - x + 40^' Ta CO p\2\2\{4x = 4x + o x ^ - x ' - x + 40 = 8V4x + + 4) < -^(2V V V 4x + 4) = x +13, d i u " = " xay o x = 65 Lai CO x ' - x ' - x + - ( x + 13) = ( j c - ) ' ( x + ) > x > - l , d o d x^-3x^ - x + > x + 13,dau"="xayra < » x = Tir X = la nghiem nhdt cua phuong trinh Ta Dat CO ^2x^-llx + ' t = •N/4X - = > X = = x - < : > x ' - l l x + 21 = V x - => x = 16 Dodo ^(t' + St' +16)-^(r' +4) + 2t-3t = 0^t'-I4t' o ( r - ) ' (/'+4/'+12r'+18/ + 24) = 0c:> -24? + 96 = (2) J * + ? ' + / ' + / + 24 = 0" Xet / = o x = Xet r ' + r ' + r ' + / + 24 = 0(3) Ta CO voi t <0 thi (2) v6 nghiem V o i f > thi (3) v6 nghiem • Vay nghiem cua phuong trinh la x = 4.Tac6 ( x ' + x - l ) ' - x ' + x ' - Dat a = x ^ + X - , taco V i x^ + xa + = o ( x ' + x - l ) ' + ( x ' + x - l ) = x'+3x +3a = x ^ + x < = > ( x - a ) ( x ^ + x a + a ^ + ) - +3= x+— V +— + > nen ta c6 a = X < : : > X ^ + X - l = X<=>X = ±l Vay nghiem cua phuong trinh la x = ±1 V i d u t o n g ho'p C h i f c n g Vi du Giai phuong trinh (2x' - 3x -18) (3x' + 2x - 27) = 4Ix^ +1 Ox' - 369x Ta CO x = khong phai la nghiem cua phuong trinh L&igidL V o i x^O, chiacahai cho x ' tac6 66 2x-32 18^ ( X —9^ 3x + 2- 27 = 41x + - 369 f -3 ^X 9^ I XJ + = 41 V X - 2? XJ + 10 Dat t = x — , ta CO phuang trinh tra /=8 (2/-3)(3/ + 2) = 4k + 10 <=>3r^-23^-8 = 0« ]_• x = -l Vai r = taco x — = x - x - = 0« x = X Vai t = - - taco x - - = - - < : > x ^ + x - = < ^ x ^ 1±V325 x 7x'-101x'+42x Vi du Giai phuang trinh = x ' - l l x + 2x^+x + 12 L&igidL Ta c6 phuang trinh tuang duang vai (2x'+ X + ) ( x ' - Ix + 6) = x ' - l O l x ' + 42x Voi X = 0, khong phai la nghiem cua phuang trinh Vai x^O, chia ca hai v6 ciia phuang trinh cho x^ ta c6 12^ x - l l + ^ ^ = 7x-101 + 42 — 2x + l + x^ V X JV ( 6) -101 ' x + — +1 x + ^ - 1 = X I XJ x+ Dat t = x + -,t > 2V6 , phuang trinh tra V (2? Xj + l ) ( / - l l ) = 7^ X «:>2/^-28/ + 90 = 0<» t = t=9 x=2 Vai / = taco x + - = 5<»x^-5x + = 0«> x = 3' X Vai t =9 taco x + - = 9<^x^-9x + = 0<^x = X 9±V57 Vay nghiem cua phuong trinh la x = 2; x = 3; x = 9±V57 Vi du Giai phuong trinh x^ - Ax^ + + Ta c6 phuong trinh tuong duong - 4JC +1 = L&igidL (x + l ) ( j c ' - x ' + x ' - x + l) = 0<» x'-5x'+Sx^-5x + \ 0' Xet phuong trinh x^ -5x^ +Sx^ -5x + \ Voi x = 0, khong phai la nghiem cua phuong trinh Voi x^O, chia ca hai ve cua phuong trinh cho x^ ta c6 + = ' x'+\-5 X x+ - + Dat >' = x + -1, taco y =x + — y =2 Ta CO - 5y + = 0<^ =3 Vai ;; = 2<»x + - = 2<»x^-2x + l = 0<=>x = l X Vai y = 3<^x + - = 3^x'-3x X +\ ^ x ^ ^ ^ ^ Vay, nghiem cua phuong trinh la x = ±1; x = 3±yf5 Vi du Giai phuong trinh 2x^ - 9x' + 20x' - 33x' + 46x* - 66x' + SOx^ - 72JC + 32 = L&igidL Voi x = 0, khong phai la nghiem cua phuong trinh Vai X ?t 0, chia ca hai ve cua phuong trinh cho x" ta c6 o r> -^rv 66 80 72 32 _ 2x - x +20x -33x + 46 + ^ — - + —- = o x ' x^ x' X 68 ^2 r„4 X + Dat y x + -,y X -9 + 20 - 3 x + — + 46 = >2>/2,tac6 ( / - / + ) - ( / -6>;) + ( / - ) - 3 ; ; + 46 = <:> / - / + / +21>;-18 = <;^[y-\){y-2){y-2){2x + 3) = Q<^y = \-,y = 2-,y^y,y = -'^ Chi CO >' = thoaman x+ —= X 3<^x^-3x +2 = 0<» x=l x=2 Vay nghiem cua phuong trinh la x = 1; x = Vi du Giai phuong trinh 2x^ - Sx" + 6x^ - 8x^ + = Lffi giau Ta co 2x' - x ' + 6x' - x ' + = «(2x +1)(x-1)' (x' + 3) = « X = Vay nghiem cua phuong trinh la x = - ^ ; x = 10 x ' - x + l- + -2 x ' - x + x ' - x + Ldi giai Dat / = 2x^ - x +1, / > 0, phuong trinh da cho trd t =2 10 = o <:>6r-6/-12 t = -\ t+6 -t + -t+2 Voi t = -\ x=l V6i ^ = 2«2x'-x + l = < i > x ' - x - l = o 1• x= — Vay nghiem cua phuong trinh la x = l;x = - ^ Vi du Giai phuong trinh Vi du Giai phuong trinh x ' + x + = + x + x ^ x'+2x +3 +2x + = - o + x + = 0(v6nghiem) Vai / = -2,tac6 + 2x + 3, ta c6 Led giaL Dat r = = / + l « / ^ - = 0<=>/ = -2;r = Vai r = 2,tac6 x^+2x + = 2<»x^+2x + l = < = > x - - l Vay nghiem cua phuang trinh la x = - Vi Giai phuang trinh x^ + 4x^ = (x-2)^ Lei giai DiSu kien x 2x \ x+x-2 Dat r = x-2 Ta c6 phuang trinh tuang duang vai ( x-2 =5 « x-2 ^ x-2 Taco / ^ - / - = 0<i> Vai / = - l < ; ^ if: =5 x-2 t = -\ t=5 = -!<:> x ^ + x - = « x=l x = -2 V a i / = < » x ^ - x + 10 = (v6 nghiem) Vay, nghiem cua phuang trinh la x = l;x = - Vi du Giai phuang trinh x + = (x-fl)^ LoigiaL DiSu kien x - Ta c6 phuang trinh tucmg ducmg vai = 3- X x+1 Dat x+1 2x^ x+1 .Taco r + / - = 0<=> Vai / = taco x - x - l = < » x = + 2-3 = x+1 /=1 t = -2,' 1±V5 Vai / = -3 ta CO x^ + 3x + = (v6 nghiem) Vay nghiem cua phuang trinh la x = 1±V5 70 Vi du 10 Giai phuang trinhx' + 9x' + 27x-54 - h \^] L&igidL Taco X'+9JC^+27X-54 = 0<:^(X + 3)^-81 = 0<»X = -3 + V8T Vay nghiem cua phuang trinh la x = -3 + ^81 ^x + 3^' 3{ + 3f ++ 5(x + 3)(x-l) = 9(x V[ du 11 Giai phuang trinh (x-1) x-1 Led giai Dieu kien x Dat a = x + Phuang trinh tra x-1 + 3a^b + 5ab^ -9b^ =0 (a + 26) +56' = ( V N ) + Vl7 x+3 Vai a = Z?<=> x - = x - l<»x^-3x-2 0<»x = + =717 Vay nghiem cua phuang trinh la x = 28x + Vi du 12 Giai phuang trinh 3x'+2 9x + x - ) = x-3 LcigidL DiSu kien x ?!: 28X + x ' + x + 3(3x'+2) + x - 3(3x^+2) ^ Ta CO 9x + = = —5^ = +1 x-3 x-3 x-3 x-3 t =\ 3x'+2 Dat Do taco /(3r + l) = 4<::>3?'+/-4 = 0<» 4t=x-3 Vai r = o 3x'+2 = <^ 3x' - x + = (v6 nghiem) x-3 ^ 3x'+2 ^ , ^ ^ -2±V58 -2±V58 Vai — <=> Vay nghiem ciia phuang= — trinh<::>9x' la x =+ x - = 0<^x = x-3 f ^( 10 ^ =5 Vi du 13 Giai phuang trinh x + 2x + ^ x + UV x + 1^ L&igidL Dieu kien x ?t - Ta c6 X + 3-x+l ^2x'+lU-l Taco JC + x^+4x-2 Dat r = ^ x+l Voi / = 2x + 9- x+l = « x^+4x-2 x+l ( x ' + x - ) + 3(x + l ) 2x^+nx-l x+l 2(x'+4x-2) x+l x+l =5 +3 t^l , t a c f(2? + 3) = c ^ / ^ + ^ - = ^ _ _ ^ ' ~ x^+Ax-2 x+l = l<=>x^ + x - = 0<=>x = -3±V2T x'+4x-2 ^ ^ , ^ -13 + 7161 Von t = — = — o x +13x + l = 0<::>x = x+l , , -3±V2T -13±Vl6T Vay phuofng trinh co nghiem x ,x = V i dv 14 Giai phuong trinh + lj yX X ' Lotigiau * ) Dieu kien: x*-l,x^ ^ + r x - 11 N3 VX + 2y -2 Dat a = x+l Ta CO a' + 4b' - 3a'b <^(a + b){a - 2bf = V a i fl + Z) = ta C O x-1 x+l Voi a - 26 = ta C O •+- x+2 ^(x-1) = 3(x + i r ( x + 2) x-1 ,b= x+2 a+6=0 a-2b^0 = 0o2x^+2x-l-0ox = 2(x-l) x+2 X+l - + V3 = < ^ - x ^ + x + = 0<z>x = l±^/3 Vay nghiem cua phuong trinh la V i du 15 Giai phuong trinh x^ - x^ +1 x^ + 2x + +— = 2- x X + X +1 X L&igidL Dieu kien x ^ 0; x^ + x +1 Ta CO phuong trinh tuong duong voi x^-x^+1 , x^+2x + l X +X + -+ x+ l+^ , x^ + x + X X + X +1 1= +— = 72 Dat / = x^+x + l , ,tac6 / + - = < = > r - / + l = 0<=>^ = l X t + X+ , , ^ X Vay nghiem cua phucmg trinh la x = -\ Bai tap tong hop Chuong ^- ' Bai Giai phuong trinh x ( x ^ - ) = V3 Bai Giai phuong trinh [x^ - 3x + 2)(x^ +15x + 56) + = Bai Giai phuong trinh x^ + 2V7x^ + 7x + V ? - = Bai Gia su cho so thuc a,h,c thoa man cac dieu kien a > 0,fee= la^, a + b + c - abc Chung minh rang a > I + 2V2 Bai Gia su phuong trinh ox^ +fex^+ cx + la x,, X j Chung minh rang x,X2 > = (a ^ O) c6 hai nghiem khac ^ 4a Bai Cho phuong trinh x** + ax^ +6x^ + cx + l = c6 nghiem Chung minh ring a^ +b^ Bai Cho 2a + 3b + 6c = Chung minh rang phuong trinh ox^ + 6x + c = c6 nghiemtren [O; 1] : ^ Bai Giai phuong trinh (x^ + x +1) (x^ + 2x^ + 7x^ + 26x + 37) = (x + 3)^ Bai Gia su a,b,c la dai ba canh cua mot tam giac vuong c6 canh huyen la a Chung minh rang phuong trinh -ax^ +bx + c = c6 hai nghiem phan biet XpXj thoa man dieu kien < X, < X2 < Bai 10 Giai phucmg trinh •^/3T25 x''-13^-fe x ' + 4!^3T25 x ' + 4VS'^3125 = 73 Bai 11 Gia su a,b\a.hai s6 thuc thoa man -3a^ +5a-2016 = va 6'-36^+56 + 2010 = Tinh a + Bai 12 Giai phuong trinh +Aax^ +6b^ +4c^ x + \ 0, a,b,c cac so thuc duong, a < biet rSng phuong trinh c6 nghiem thuc Bai 13 Giai phuong trinh 64x' - 96x' + 36x' - = Bai 14 Giai phuong trinh - 15x^ + 45jc - 27 = Bai 15 Giai phuong trinh x+ x+2 Bai 16 Giai phuong trinh 4x^ - 3x^ + 3x - = Hirang dan bai tap tong hgrp Chufomg l.Taco Jc(x'-2) = V3 <=>x'-2x-V3=0 « ( X - N / ) ( X ' + > / X + I) = 0« x-42=0 = V3 + \ <=>x x^+^x Vay nghiem cua phuong trinh la x = V3 Phuong trinh tuong duong voi ( x - l ) ( x - ) ( x + 7)(x + 8) + = o [(x - l)(x + 7)][(x - 2)(x + 8)] + = in x^+6x-7 x'+6x-16 + 8= «(x'+6x) - ( x ' + x ) + 120 = <^(x^+6x-15)(x'+6x-8) = o x = -3±2V6;x = - + 717 Vay nghiem cua hk phuong trinh lax = -3±2V6;x = -3± Vn 3.Tac6 x^ + 2>/7x^+7x + V - l = <::>x(x + V ) \ V - l = 0<=>x|(x + V7)^-lJ + x + V - l = <=>x(x + >/7-l)(x + V7+l) + x + >/7-l = / : 74 O ( A ; + V - ) ( X ' + ( V + I ) X + I' -(V7 + l ) ± V + 2V7 X = Ta C Ofee= 2(3^, + c = a c - a = a ' - f l Do b,c la hai nghiem ciia phuomg trinh x^ - (2a' - a ) x + 2a^ = Ta phai C O A = ( a ' - a)^ - 8a^ = Do > I+2V2 ( a ' - 4a^ - 7) > , ket hop voi a > ta C O a > I +2V2 V V i X p X j la hai nghiem ciia phuomg trinh da cho nen ta c6 ax\ bx^ + cx, + <i = 0, axl+bxl Tru theo v l ta C O a(x,'-X2) + + cx2+d fe(xf-X2) =0 + c(x,-xJ^O V i X, ^ X2 nen ta C O a ( x , + X ) ^ + ( x , + X ) + c - a x , X = Do X, + X2 la hai nghiem ciia phuomg trinh at^ + bt + c- ax^x^ = D I phuomg trinh c6 nghiem ta c6 A = 6^ - 4ac + Aa^x^x^ > • » X1X2 > 4ac-6' 4a^ Ta C O dieu phai chung minh Gia su phuomg trinh c6 nghiem XQ, XQ ?t 0, ta C6 Xo+aXo+feXo+cXo+l =O o = - X o + ^ - a X o - c - ^ ^0 Ta C O {a'+b'+c') ^ ^ Xo'+ — + -1 -^0 + — ^2 -aXo-c- X o ' + ^ +l \2 aXo+ Xo + — - a x p - c V ^0 + — ^0 y Suyra [a^+b^+c^) ^ + ^ ^0 ^0 + 0y _ voi ? + r+1 + — > r - 2)(3^ + 2) > 2luon dung vi r > Mat khac j — ^ > - '- 4/ '- > Vay + 6^ + > - , dau bang xay rakhi a = b = c = — ung voi = 1, 3 2 hoac a = c = —,6 = -— ung vai x^ = - Dat P(x) = <3x^ + ZJX + c, ta CO P{x) la da thuc xac dinh tren M Taco P(0) = c;P(X) = a + b + c; AP = a + 2Z) + 4c.Ta c6 P(0) + P(1) + 4P v2y = 0, do cac s6 P(0); P(l); P CO It nhat mot so khong am, mot s6 khong duomg Do P(x) = c6 nghiem tren [O; / oN Cach khac: Xet P(0).P = - c ' < v3 Ta CO (x^ + X + l)(x' + 2x^ + 7x^ + 26x + 3?) = 5(x + 3)^ o ( x ' + x + l ) | ( x ' + x + l ) ' + ( x + 3)'J = 5(x + 3)' Dat a = x^ + X +1, = X + Ta CO a [a^ + 4Z>^) = 56^ <^a^ + Aab^ - 56^ = x/-> ^ \ —b = Q <:>{a-b)ia'+5ab + 5b') = Q<^ , , Vai a = < ^ x ^ + x + l = x + o x ^ = < = > x = ±V2 76 Vai a ' + f l + Z ) ' = « 2^ V Do + X +1 J 0, v6 nghiem Vay nghiem cua phuong trinh la x = +V2 Ta C O a, c>0 nen -ac>0, do phuong trinh -ax^ +bx + c = c6 hai nghiem XpXj trai dau Gia su x, < < X2 Ta C O - o x f + Z J X , + c = <=> oxf = Z>x, + c Do a'x^ = (bx, + c)' < + )(xf +1) = a' (xf + ) Suy x,' < xf +1 xf - < « - >/2 < X, < ^ x," < xf + « ( x f +1) (xf - 2) < V2 Tuong tu -V2 < Xj < V2 Vay ta c6 -V2 < x, < Xj < V2 10 V o i X = khong phai la nghiem cua phuong trinh A u ^ u ^ 4^5 13'-^ Vol X 7t 0, phuong trinh tuong duong voi x + — + —— = —1= X X v3125 Ap dung bat dang thuc Co si cho 13 s6 duong ta c6 x" X* x' •" "• V5 x' x' Vay nghiem cua phuong trinh la x = n V5^13^-fe •" Vs Dau " = " xay <» — = — = ^ <^ X = ^ x' x' ^'i x' !^^T25 ll.Dat / ( x ) = x ' - x ' + x = ( x - l ) ' + ( x - l ) + 3.Tac6 / ( a ) = ; / ( ) = -2010 Taco / ( a ) - = ( f l - l ) ' + ( a - l ) ; f{b)-3 Suyra f{a) +f{b)-6 ^ = (a-\f c^(a = {a-\f +{b-\f + b-2){{a-\f +2(a-l) = (b-\f + {b-iy +2{b-\) +2{b-l) +2{a + b-2) -{a-\){b-l) + (b-\f +2] = 77 =0 <:>a + b-2 = 0<:>a + b = Vay a + b = 12 Do cac he so cua phuong trinh dku duong va phuong trinh c6 nghiem thuc Xj,x^,x^,x^ thi ca nghiem dSu am Dodo >0 Theo dinh ly Viet ta c6 {-x,) + {-x,) + {-x,) + {-x,) \{-x,){-x,){-x,){-x,) =\ Su dung bat dang thuc Co-si cho s6 -x,, - ^2, - X3, 4fl = ( - X , ) + ( - X ) + (-X3) + ( - X , ) > 4/(-x,) -x^,-x^,-x^,-x^ = 4a ' X - f>:f: > 0, ta c6 ) (-X3) ( - X , ) = Do a = l va ta suy (-x,) = (-x^) - (-x,) = (-X4) = hay cac nghiem cua phuong trinh la x, = Xj = X3 = x^ = - 13 D a t / ( x ) = x ' - x ' + x ' - T a c / ( x ) la mot ham da thuc va la ham chan tren R Ta chung minh / ( x ) = c6 nghiem phan biet thuoc (-l;l) Do / ( x ) la ham chSn tren R nen ta se chung minh / ( x ) = c6 nghiem thuoc (0;1) That vay f(0).f <0;/ r 1\ <0;/ T \ ./(1)<0 nen / ( x ) = c6 nghiem thuoc (O; l) Dat X = cosr voi te(0;n:) phuong trinh tren tro 64 cos* t - 96 cos^ t + 36 cos^ / - = Ta CO cos6r = 4cos^ 2?-3cos2? = 4(2cos^/ -l)^-3(2cos^?-l) = 32 cos* / - 48 cos'r +18 cos^ ? - Do phuong trinh tren tuong duong voi ' '' - 2cos6?-l = 0<::>cos6/ = -<»6^ = ± —+ A:2;r<:> ? = + — 18 78 'n Do t e (0; TT) nen t Sn In \\n Xlin \ln Tir phuomg trinh c6 nghiem la 5;77;T ll;r 13;r 17:7r cos—;cos—;cos—;cos :cos :cos 18 18 18 18 18 18 14 Dat x=i4^t thay vao phuang trinh ta c6 288V3r' - 350V3r' + 90V3f - 27 = * < » ( l r ' - r ' + ^ ) = >/3 Ta tim nghiem t e [ - l ; l ] ciia phuang trinh Dat r = c o s a , a € [ ; ; T ] Taco cos 5a = 16cos'Qr-20cos^a + 5cosa Phucmg trinh tra cos 5a = — = +—+ , v i a e fO; n\n 30 ^ •' n Un: Sn Un: Thn [30 30 30 30 Do phuang trinh bac c6 nhigu nhdt nghiem nen phuang trinh c6 nghi?m la cos—; cos ;cos—;cos ;cos 30 30 30 30 x+1 15 Dieukien x ; t - D a t a = ; = x + 2, Phuang trinh tra x+2 a' + la^b-3b' ^(a-b) = 0^{a-b) a +—b ( a ' + 3ab + 36') = = x+l <^a = b<^ x+2 = x +2 x ' + 3x + = 0(v6 nghiem) Vay phuang trinh v6 nghiem 16 Ta CO 4x' - 3x' + 3x - = <i> 3x' + (x - ) ' = <:> 3x' = ( l - x)' <:^^X^\-XO X = Vay nghiem ciia phuang trinh la x = 1+ ^ " 79 ChUctng PHtfdNG TRINH, BAT PHlTdNG TRINH V T I A M o t s d a n g p h i p c n g t r i n h c c ban Chii y: G day chung toi chi dua cdc dang ca ban cho phucmg trinh, vai bat phucmg trinh cung cd the cd dang tucmg tu Chung ta can luu y rang, gidi bat phucmg trinh, chdng hgn f (x) > duang I6i chung la di tim nghiem cua phuomg trinh / ( x ) = 0, sau xet ddu cua f (x) tren moi khodng duac tqo thdnh bai cdc nghiem cua phucmg trinh f (x) = ,,tu de lay nghiem cua bat phucmg trinh f (x) > V ^ = V K ^ Cdch gidi: Binh phuang hai ve, phucmg trinh tren tra M ( X ) = v(x), Icem theo dieu kien hoac u{x) > hoac v(x) > ^4^ = ^^ Cdch gidi: Lap phuang hai ve, phuang trinh tren tuang duang vai w(x) = v(x) ^/w(x) =v(x) Cdch gidi: Dat diSu kien v(x) > 0, binh phuang hai v l , phuang trinh tren tra M(X) = V^(X) 3/^^00 ^ =v(x) Cdch gidi: Lap phuang hai ve, phuang trinh tren tuong duang vai uix) = v\x) Va + x + yjb-x + c^J(a + x)(b-x) =d Cdch gidi: Dat di^u kien x>-a,x<b r6i dat V x T a + yjb-x = t, bieu dien yl{x + a){b- x) theo t Khi phuang trinh da cho tra mot phuang trinh bac hai theo t 80 au(x) + bv(x) + Cy]u(x)v{x) = Cdch giai: Dat dieu kien u{x)v{x) > Xet cac trucmg hop theo diu cua v(x), sau chia ca hai ve ciia phuang trinh cho v(x), roi dat u{x) v(x) Khi phuang trinh da cho tra mot phuang trinh bac hai theo t , ^7 <3|x + Vfe-jc^ J + cxylb~x^ +d = Cdch giai: Dieu kien x e {-4b;4b] Dat x + yjb-x^ =t,rh\ diln xylb-x'^ theo t Khi phuang trinh da cho tra mot phuang trinh bac hai theo t x^+b + d = a x + 4x^ +b + CX Cdch giai: Dat x + Vx^ + = ?, r6i biSu dien x x + ylx^+b^ theo t Khi phuang trinh da cho tro mot phuang trinh bac hai theo / u^(x) + aylb-au{x)=b b > u{x) Dat Jb-au{x) = ?, t > d6 dua vh he d6i Cdch giai: Qiku kien — a a ' u (x) + at=^b xung theo / va u{x) Do la he [t^ +au(x) = b Ghi chu: Phuang trinh dang c6 the tong quat hoa dang u"(x) + a!^b-au(x) - b hoac u"{x) + b = a^au(x)-b vai each giai tuang tu 10 ifii^ - ifi^ = v(x) - uix) Cdch giai: So sanh u(x) vai v(x) Gia su u{x) > v{x) Suyra ^u{x) > ^v(x) => v(x) > u(x) Do phuang trinh da cho tuang duang vai u(x) = v(x) ' Ghi chii: Phuang trinh dang c6 the tong quat hoa dang -A ^ !(u(x) - !fix) = [v(x) - uix)] A(x), ^(x) > 81 B Phu'cng phap giai 1) Cac phucmg phap dai so 1.1 Phwffng phap nang len lily thira Nhan xet: Trtrac ndng len luy thira chung ta nen sdp xip lai cac s6 hang a hai ve de sau ndng len lily thira an x nam a ngodi can thuc triet tieu hoac CO bdc thdp nhdt, dong thai luu y den dieu kien de hai vi cua phucmg trinh khong dm (doi vai viec ndng len lOy thica bdc chdn) Vi du Giai phucmg trinh VJC-1 + Vx + Jlx = L&igidL *) Dieu kien: x > Phuang trinh da cho tuomg duong vai f f \ x = -\ x=2 Doi chieu dieu kien ta dugc nghiem cua phucmg trinh da cho x = Vi du Giai phuang trinh 4-x + - / - ^ - - V x + T = f-l<x<0 L&igiaL *) Dieu kien: -X-ylx + \>Q Khi phuang trinh da cho tuang duang vai yJ-X-y[x + \ - V - X <=>-X- ^jx + l = - X - 2yf^ + X +1 + 2y]x + \ - x <=> + V x T T = 2V X -l<x< — 16 o - - x = 2Vx + l (thoa man) = <=>x 25 + 25x^+20x-4(x + l) 1 Vi du Giai phuang trinh Jx — + - - =x X x >0 X L&igiaL *)Di6ukien: l >0 X x^O o x>l -l<x<0 R5rang, tirpucmgtrinh da cho tac6 x>0 Suy x > Khi phuong trinh da cho tuong duong vai X \ -x-2yf^ X x+l \ = 1^ X 0<^ = n , l + >/5 , , <=>Vx - x = l < = > x = , thoa man Vi du Giai phuomg trinh >j2x-\ 3x-x'^ L&igidL *) Dieu Icien: x > — ^ Khi phuomg trinh da cho tuomg duorng vai -\ 3x-x^>0 -l + x2 x=\ 3x-x^>0 ^ x - l = ( - l + 3x-x^) x = 2-y[2 (jc-l)^(x^-4x + ) - V i du Giai phuomg trinh ^jx + + VSxTT = 2^fx + -j2x + LoigidL *) Dieu Icien: x>0 Phuomg trinh da cho tuomg duorng vai V3x + l - V x + = > / ^ - V ^ ^ x + 3-2V3x + W x + =5x + 3-4V^V7+3 o V3x + W2x + =2VxVx + <=>6x^+8x + = x ^ + x < » x = l Thay vao phuomg trinh da cho thoa man ' ' , V i du Giai phuomg trinh V - V l O ^ ^ = x - L&igidL *) Dieu kien: 4-3N/10-3X>0 10-3X>0 74 10 <=> — < x < — 27 83 •1 Phucmg trinh da cho tuong duong voi - V l O - x = x ^ - x + c^Ax-x^ =3yl\0-3x <=>16x^-8x^+x'^ =9(10-3x) < ^ ( x - ) ( x + 2)(x^-7x + 15) = "x = - _x = Doi chiSu dieu lei en ta dugrc nghiem cua phuong trinh da cho x = Vi du Giai phuong trinh + = ^5x + \ L&i gidL Lap phuong hai ve cua phuong trinh da cho ta dugc x - + f c - l f e - l ( ^ x - l + f c - l ) = 5x + l => ^ x - l f e - ^ x + l = x-0 <»30x^-19x^ = o 19 X= 30 Thay vao phuong trinh da cho dugc nghiem x = 19 30 Vi du Giai bat phuong trinh - 3x + >0 i2/3x^+2x + l L&i gidL Bat phuong trinh da cho tuong duong voi A / ? + 2x + l >3x + (1) a) X < -—, (1) thoa man b) X > -—, binh phucmg hai ve ciia (1) ta dugc ^=^^<x<=^^ 3x2+2x + l>9x2+12x + Suy «> —2 < x <^ - + V7 Ket hgp ta dugc nghiem cua bat phuong trinh x < -5 + V7 6 84 V i du Giai bat phucmg trinh V^^^^ }^) + ^ ^ T ^ > J L ^ ylx-3 Vx-3 (Di thi DH khSi A 2004) L&igidL *) Dieu kien: x2-4>0 <::>X>4 x>3 B i t phuomg trinh da cho tuong duomg vai y]2{x^ -16) + x - > - x « y j l i x ^ -16) > - x +) Voi X > 5, thoa man +) V o i < X < 5, binh phuong hai ve cua bat phuomg trinh tren ta dugc 2(x^ - ) > 100 - 40x + 4x^ o x^ - 20x + 66 < =^ 10 - V34 < X < Ket hop ta dugc nghiem cua bat phuomg trinh: x > 10 - ^34 V i du 10 Giai bat phuong trinh V x - - > V2x-4 (Di thi DH khSi A 2005) L&igidL *) Q\eu kien: x > Bat phuomg trinh da cho tuong duomg voi V x - > V x ^ + V x - < : > x - l > x - + 2^yx^.^/2x-4 < » x + > V x - l V x - <^x^ + 4x + 4>2x^ - 6x + <::::>X^-10X<0^2<X<10 Bai tap phan 1.1 Bai Giai phuong trinh Vx + - Vl - x = V l - x Bai Giai phuomg trinh -^x-l^x^ + Vx + V x ^ = — ^ X -1 Bai Giai phuong trinh V4x + V ? T l = x + ; :; Bai Giai phuomg trinh A/X + + l]x-\ >/5x Bai Giai bdt phuomg trinh \/4x^ - x + - 4x + > 85 Bai Giai b i t phuong trinh V2X + 1+1 5x + Bai Giai phiromg trinh ^J2x^ +2x + + yjlx^ +2 = VSx^ + 2x + \ Vx^ +6 Huong dan bai tap phan 1.1 Dieu kien: - < x < — Vigt phuong trinh dang + = -Jl-x + ^Jl-2x Binh phuong hai ve, nit gon dugc 2x + l = V r ^ V l - x <»- <^x^0 x ^ + x + l = l - x + 2x^ Dieu kien: x > Phuong trinh da cho tuong duong voi x-1 De pha tra tri tuyet doi xet hai truong hop: + ) X > , truong hop khong c6 nghiem +) < X < , truong hop c6 nghiem ^ - ^ • Dieu kien: 4x + \/9x^ +16 > Phuomg trinh da cho tuong duong voi 'x>-2 x =0 x = ±l 4x + V x ^ + = x ^ + x + 4 ^ +1 + yjx-1 = yfSx Lap phuong hai ve ta dugc :kt^ i x + ^ / ^ ^ / x ^ ( ^ / ^ + ^ / ^ ) = 5x => ^ / x T T ^ / ] ^ ^ = X Giai phuong trinh nay, thu lai dugc nghiem x = 86 Bat phuang trinh tuong duong vai ^IAX^ -6X + >4X-3 3 Xet hai truong hop jc < — va x>— duac nghiem ciiabat phuang trinh x < 4 ^ h • -••f Dieu Icien: X > — , x ^ — Xet hai truong hop: ' t i- vg; i +) -— < X < -—, bat phuong trinh thoa man * 5x2 + l > V2x + <=> +) x> — , bat phuong trinh25x2+10x tuong duong + l <vai 2x + l 25 ^x>0 Ket hop lai dugc nghiem — < x < — ; x > "2/ Binh phuong hai ve cua phuang trinh ta dugc V2x^ +2x + 5.V2x^ +2 = V3x^ +2x + l.Vx^ +6 « (2x' + 2x + 5)(2x' + 2) = (3x' + 2x +1)(x' + 6) <::>4x'' +4x^ +14x^ +4x + 10 = 3x' +2x' +19x^ +12x + « ( x ' - 5x'+ 4) + ( x ' - 8x) = fx = ±2 o f x ^ - ) ( x ' + x - l ) = 0« ^ ^ [x = - l ± 1.2 Phuffng phdp sie dung bieu thuc Hen hffp Vi du Giai phuong trinh Vx^ +x + l + Vx^ - x + = lo-/^/fl/ Tu phuong trinh ta c6 x = Vx + x +1 - V x ^ - x + l Vx^ ++ XX ++ 11 - Vx^ xx ++ 11 == 2X Nhu Suy ravay ta dugc +x + l = <=> x = ^^ 87 V i du Giai phuong trinh +X +6 + +x + = x + X L&igidL *) DiSu kien: x ^ Nhan lien hcrp voi trai ta dugc +4 = ^sllx^ +x + 6-^]x^ +X + x^+4 X Hay yllx^ +x + 6-ylx^ +X + = x (1) Ket hop (1) voi phuong trinh da cho dugc Vx^ + X + = - <^x'^ + +2x-4 = 0,x>0 X ( x - l ) ( x ^ + x + x + 4) = , x > « x = l Vi du Giai phuong trinh A/3X' - x + - V x ' -2^^|3x- - x - l - V x ' -3x + 3x^-7x + 3>0 L&igidL *) Dieu kien: x^-2>0 3x^-5x-l>0 x^-3x + > Phuong trinh da cho tro V x ^ - x + - V x - x - l + V x ^ - x + - V x - =0 <^ 2x-4 3x-6 , , + ^ , =0 V3x^-7x + + V x ^ - x - l Vx^-3x + + V x ^ - (x-2) ,V3x^-7x + + V x - x - l \/x^-3x + + V x ^ - y X = 2, thoa man V i du Giai phuong trinh V^Thl - V ^ = V2 ( - ^3^) L&i giai *) Dieu kien: - < x < Nhan voi lien hgp cua bieu thuc can ve phai ta dugc ( V ^ - V ^ ) ( V ^ +V ^ ) =V ( V ^ - V ^ ) -V 88 +) V + T - V ^ = 0<:^X = +) ^J7+l + ^ / ^ = y[2 Binhphuonghai ve ta duoc V I T T + V ^ + ^ x ^ ^ = + Vs^)^ Chuyrang S.uy + (1) =4+ 2V^.V3^>4 +2 > Dfiu dang thuc xay x = - , x = Do phuong trinh (1) c6 nghiem x = - l,x = Tom lai, phuong trinh da cho c6 nghiem x = l , x = - l , x = Vi du Giai bat phuong trinh +Vx-1 < V3x-2 V4x + LMgiaL *) Digu kien: x > Khi b i t phuong trinh da cho tuong duong 2x-l <=> , < V3x-2-V^ V4x + 2x-l 2x-l V4x + V3x-2+V^ < » V x - + V x ^ < V4x + (dox>l) « x - + V ( x - ) ( x - l ) <4x + l « V ( ^ - ) ( x - l ) < <:^3x^-5x-2<0« <x<2 • Ket hop dieu kien, ta c6 nghiem cua bat phuong trinh la < x < Vi du Giai bat phuong trinh ^ , > V2x + Vx+l-Vx-2 TA.* M ^ - ' I - " fVx + - V x - ^ Lai giau *) Dieu kien: < o x>2 [x>2 Nhan lien hop voi mau so cua ve phai ta dugc Vx+T + > / x - > > / x + o x - l + x / ^ V x - > x + <:> Vx + l V x - > « x ^ - x - > < ^ " x < - x>3 • Doi chieu dieu kien ta duoc nghiem cua bdt phuong trinh V i du Giai b i t phuang trinh ^^5—^ + x>2 > V3x-2 L&igidL *) Dieu kien: ^ - • Bat phuong trinh da cho tuong duong voi Vx + Vx + v3x-2+Vx Ro rang x = khong thoa man (1) Xet hai truang hop sau: a) — < X < K h i (1) tuong duong voi ^ < , ^ ^<=>V3x-2+Vx <Vx + Vx + V x - +Vx <z>4x-2 + V x - V x <x + < » V x - V x < - x / O ( x ^ - x ) < - x + 9x2 <:>3x^+22x-25<0 •o < X < Suy — < X < b) X > I Khi (1) tuong duong voi > yJx + , p^c:> V x - + V x >Vx + V3x-2+Vx Giai tuong tu truong hop tren ta dugc nghiem x > Vay, bat phuong trinh da cho c6 nghiem — < x < hoac x > Vi du Giai b i t phuong trinh 2x + Vx + > V3x +1 LdigidL *) Dieu kien: x > B i t phuong trinh da cho tro 2X-1>V3X-VJC+TO2X-1> ^/3^ + V ^ • 90 Xet cac truong hop sau: +) ^ - ~ ' thoa man +) 0<x <^,hk phuong trinh tro 1< - ^ = - ^ - = o V3x + Vx +1 <::> 4x + + lSx.y[x Vs]^+v^m < +\ 2x + V B X V X + 1< 0O X= + ) X > - , bdt phuong trinh tro > ;— ^ , - = <=> VSx + Vx + > 1, thoa man voi moi x > — Tom lai, bat phuong trinh da cho c6 nghiem x = 0, x > i „, Vi du Giai bdt phuong trinh 2x>/x + + V3x + < Vx + Lo*/ ^ifl/ *) Dieu kien: x > - — Bat phuong trinh da cho tuong duong voi •l - x 2x^/x^-3 - V x + < Vx + - V x + <^(2x-l)>/x + < o(2x-l) + Vx + + V3x + J <0<:=> — < x < Bai tap ph'an 1.2 Bai Giai phuong trinh V'2x^ + x + l + Vx^ - x + = 3x Bai Giai phuong trinh V2x + - V ^ Bai Giai phuong trinh .^ Vx + + VxTT Vx + + V3x + V3x + V3x + Bai Giai phuong trinh yjlx^ +x + + yjlx^-x + l =x + Bai Giai bat phuong trinh yllx + Bai Giai bdt phuong trinh Bai Giai hk phuong trinh yj\ x - yl\-x >/4x + l <x H u a n g dan bai tap phan 1.Z T u phuong trinh da cho ta c6 x > Ro rang x = khong thoa man phuong trinh da cho, nen x>0 Nhan voi lien hop cua ve trai ta dugc ^J2x^ +x + \ A / X ^ ^ - ^ T T = Ket hop voi phuong trinh da cho ta c6 Giai phuong trinh dugc nghiem x - + X+ — • 5x + l -19 + 3V65 14 0<x^l <^0<x^l Dieu kien: >/2x + l - V ^ ^ Nhan voi lien hgp mau so cua ve trai ta dugc ^J2x + \+^I3^ = 1-x yl3x + _ PTTl, FT rr~7^ <=> V2x + +V3x = V3x + 1-x < » x + l + 2V2x + l V ^ = 3x + o V x + l.V3l = l - x 0<x<-4 + ^[2\ <:> X = 10 4(6x2+3x) = l - x + 4x^ Dieuki^n: x > - - Phuong trinh da cho tuong duong vod = V3x + l - V x + T yfI+2 92 Nhan voi lien hop ciia ve phai ta dugc Ix x=0 <=> -2 + 24l Giai dugc nghiem ciia phuong trinh x = 0, jc = T u phuong trinh da cho ta c6 x > - Nhan lien hgp yk trai ta dugc 2(x + 4) = (x + 4) ^Ix^ + x + - V2x^ - x + o = V2x2+x + 9-V2x2-x + l K6t hgp voi phuong trinh ban d i u ta dugc l^llx^ +X + = X + Giai phuong trinh dugc nghiem x , V2x + 0, x = — +1 < V2x + Digu kien: x > - - • Bat phuong trinh da cho tuong duong voi 2x 2x <V2x+l-lC:> V2x + V2x + 2x V2x + l + r Xet cac truong hgp sau: + ) X = 0, thoa man +) < X < 0, bat phuong trinh tro V2x + c^V2x + l+l>>/2x + V2X + 1+1 o 2x +1 + 2V2X+T +1 > 2x + 2V2X+T > Truong hgp bat phuong trinh khong c6 nghiem + ) X > K h i bat phuong trinh tro , ^ <^J=—<i>V2^ + l<V2^^ V2x + V2X + + Truong hgp giai dugc nghiem x > 45 93 45 Vay, bat phuomg trinh da cho c6 nghiem x = 0, x > —• DiSukien: xTu- u 0<x<3,x^\ ~ - ' J Nhan lien hap vai mau so cua ve trai ta duac V2x + V - X > x-l V4x + x-\ , « Xet hai tuong hap sau: +) < X < 1, bat phuomg trinh tra V2X + V - X < V4x + <=>2x + - x + V x V - x <4x + l <=> V X V - X < 3x - <=> • x> 4(6x-2x^)<9x^-12x + Truang hop b4t phuang trinh khong c6 nghiem + ) X > 1,bat phuang trinh tra thanh-Jlx + \ / - x > V4xTT Truang hap giai dugc nghiem X > Dieu kien: -1 < x < < Nhan lien hgp vai v l trai ta dugc 2x < x^VTTx + >/l-x j , Xet cac truang hgp sau: + ) X = 0, thoa man +) -1 < X < 0, bat phuang trinh tra < V l + x + V i P x c > < + V l - x ^ <::>l<Vl-x^ , thoa man +) < X < 1, bat phuang trinh tra > Vl + X + Vl - X o > A/I - x^ , khong thoa man Tom lai, bat phuang trinh da cho c6 nghiem -1 < x < ' +-S <* 1.3 Phucmg phdp phan tick nhan tie Chii y: Trong nhieu truong hgp viec phdn tick nhan tu dugc dinh huang qua du dodn mot nghiem cua phuomg trinh Dieu dua tren ca sa: niu / ( x ) = CO nghiem Id x^ thi c6 the phdn tich / ( x ) = (x - x^ )yj (x), miin Id c6 su them - bat, tdch - ghep so hang phii hgp Vi du Giai phuomg trinh (x - 2)V2xTT + 3x^x^+2 L&i gidL *) DiSu kien: ^ ^ - ^ • Phucmg trinh da cho tuomg duong voi (x-2)V2x + l = ( x - l ) ( j c - ) « x=2 x=2 V2x + x=4 Vi du Giai phuong trinh Vx^ - 2x + Vx^ - x = x^-2x>0 +3x "x<-3 L ^ - ^ w i *)Digulcien: - x=0 x>2 + 3x > Nhan thay x = thoa man phuong trinh da cho Xet cac truong hgfp sau day: a) X > Phuong trinh da cho tro - • ^ x ( x - ) + x ( x - l ) = 7x(x + 3) < : ^ V x - + > / x - l = Vx + o x - + A - V x ^ = x + 3<i>2Vx-2>/x-l = - x o ( x ^ - x + 2) = 36-12x + x2 <^x = ^ ^ b) X < - Phuong trinh da cho tro ^ x ( x - ) + ^ x ( x - l ) = ^x(x + 3) V - X + V l - x = yJ-x-3 , v6 nghiem Vay, phuong trinh da cho c6 nghiem x = va x = — - — Vi du Giai phuong trinh Vx + + 2xV2x + l = 2x + V2x^ +7x + 2x+l>0 L&lgidL *) Qihu kien: x + 3>0 2x^+7x + > <=>x> — Phuong trinh da cho tuong duong voi 2X(N/2X + - ) = (>/2x +1 -1) 95 2x = y/x + ^2x +l - \0 [x = l lx = V i du Giai phuomg trinh 2x^Jx + - Sx^ L&igidL *) Dieu kien: ~x-3 x>-3 Phuong trinh da cho tuomg duong vai (Sx^ - x y l x + 3) + ( x ^ ^ - X - ) = (4x + VJT3) (2x - V x + 3) = 4x + V x + =0 2x-Vx +3 = I-V193 X = 32 x=l V i du Giai phuong trinh Vx^ +4x + + ^Jx^ +x = V3x^ +4x + l x^ + 4x + 3>0 L&igidL *) Dieu kien: X^ + X > 3x^+4x + l > x>0 x=-l x<-3 Phuong trinh da cho tuong duong vai V(x + l)(x + 3) + Vx(x +1) = V(x + l)(3x +1) R6 rang x = - thoa man +) V a i X> , phuong trinh tren tra V x + +yfx^ V3x + Giai phuong trinh dugc nghiem x = ^ ' ^ ^ ^ ^ +) V o i X < - , phuong trinh tren tro V - x - + = V-3x- Truong hop khong c6 nghiem Vay, phuong trinh da cho c6 nghiem x = - , x = + 2719 96 V i d u Giai phuang trinh -Jx + y/lx Lai giai *) DiSu kien: x > Phuang trinh da cho tuang duang vai ( V ^ - ) + (V2JC + - ) = x2 - x - « ^ - | i : = i L = ,x.l)(.-4) Vx+2 V2X + + + ) X = , thoa man +) x^A, phuang trinh tra y[^ + Chu y rang voi x > ta c6 = x+l V2X + + V^ + V2X + 1+3 < - +- = ! <x + l 2 Dau dang thuc xay x = Vay, phuang trinh da cho c6 nghiem x = 0, x = • V i d u Giai hk phuang trinh Vx^ - x + Vx^ - x > ylx^+3x X Led giai *) Dieu kien: -x>0 x>2 X^-2x>0<::> x = x^ + 3x > x<-3 Ro rang x = thoa man bat phuang trinh da cho + ) X > , bat phuang trinh tren tra V x - + V x - > +3 O X - + 2VXM"VX-2 >x + V x ^ V x - > - x Giai bat phuang trinh dugc nghiem x > + ) X < - , bSt phuang trinh da cho tra > / l - x + yjl-x > yJ-3 2\/x^ - x + > X - 6, thoa man 2V2T Vay, bat phuang trinh da cho c6 nghiem x < -3, x = 0, x > Vi du Giai hk phuong trinh (x +1)^ < (x + 5)(l - V2x + 3)^ LMgiaL *) Dieu kien: x>-— Bat phuong trinh tro 2(x + l f (l + V2x + ) % ( x + 5).4(x + l f ~ , <»(l + V2x + 3)^ <2x + 10,X9t-l<=> V2x + < , x ^ - l — <x<0 x;^-l 2Vx + > + Vx^ + x - Vi du Giai bat phuong trinh x V x - + ^^^^"^ L&igidL *)Di6ukien: x + 3>0,x^0 x2+2x-3>0 X x = -3 x>l Nhan thdy x = - khong thoa man Xet x > Bat phuong trinh da cho tuong duong voi x V J ^ - V(x - l)(x + 3) + ^ ^ ^ ^ - > <=>(Vx + - x ) (1) >0 Nhan thay x = 1, (1) thoa man Voi X > ta CO Vx + - 2x < Do (1) tuong duong voi V7^<0<=>- X>1 x^-x^-4>0 X X>1 ( x - ) ( x ^ + x + ) > <::>X>2 Vay, b4t phuong trinh da cho c6 nghiem x = l,x > 98 Vi du 10 Giai hk phirang trinh Vx^ +12 + > 3x + Vx^ +5 L&igidL Bat phuang trinh da cho tuang ducmg voi (Vx' <=> x^ - > ( x - ) + x^ - Vx' + + Vx' + + x +2 x+2 >0 Vx'+12+4 yJx'+5+3 - Chu y ring Tucmg tir, +12 - 4] > 3(x - 2) + (Vx' + - ) x+2 Vx'+12+4 x+2 Vx'+12+4 x +2 x' + + Tir ta c6 x + Vx' + + Do (1) « X - < .,fr: (1) x+ |jc+4 <1 x <1 x + -3<0,Vx Vx' + + X < Vay, bat phuang trinh da cho c6 nghiem la (-00; 2] Bai tap phan 1.3 Bai Giai phuang trinh Vx^ - x + Vx^ + x = 2x Bai Giai phuang trinh 2Vx =2 + ^|\-^fx + Vl-x Bai Giai phuang trinh xVx + + Vx^ - x + = x + Bai Giai phuang trinh x Il + — + — Vx + - l + JX + \ X + ) Bai Giai bat phuang trinh Bai Giai bit phuang trinh 2(x + l)Vx + > 8x^ +15x + Bai Giai bit phuang trinh Vx + + x^ + x < + V3x + 99 Huong dan bai tap phan 1.3 "jc>l Dieu kien: x = x<-2 Phuong trinh da cho tucmg duong voi ^x{x +1) + •slx{x + 2) = 2x R6 rang x = thoa man Sau Ian lugt xet x > l,x < -2 Giai phuong trinh tung truong hop deu khong c6 nghiem Vay, phuong trinh chi c6 nghiem x = D i k kien: < x < Dat N/X r,0 < ? < Khi phuong trinh da cho tro [i^K^t + VrT7(i + V i + r ^ ) ] =0 r = Tu thu dugc nghiem cua phuong trinh x = Dieu kien: x > - Phuong trinh da cho tuong duong voi ( Vx^+-1-1 l ) ( ^V / x ^ - x + l - x = Giai dugc nghiem x = 0,x = 'x + Dieu kien: >0 X -2<x<-l <^ x>0 ,x>-2 Phuong trinh da cho tuong duong voi X X x +1 ( V ^ - x ) = Giai dugc nghiem x = 2, x = ——^ Bat phuong trinh CO nghiem X = l,x > Dieu kien: x > - Bat phuong trinh da cho tuong duong voi 2(x+i)V^+4>8(x+i)^-(x+4) 4(x +1) + Vx + 4l.r2(x +1) - Vx + <0 100 4(x + l) + VxT4 >0 2(x + l)-Vx + <0 32 -4 < X < <=> 32 <x<0 x>-3 Dieu kien: - x> o x> 31 Bit phiromg trinli da cho tra +x-2<V3x + l-Vx +3 2(x-l) «(x-l)(x + 2)< V3x+T + Vx + < (1) x+ 2V3x + +Vx + 3, Voi x > - - taco x + > - > V2 va V3x + +Vx + > V2, nen 3 2 x+2 — ^ ^ = > V - — = V3x + l+Vx + V2 Do bat phuong trinh (1) tuonng duong vai <» — < x < l x-l<0 ^ Vay, bit phuomg trinh da cho c6 nghiem la: - ^ < x < 1.4 Phuongphdp dat &n phu hoan toan ' a) Dat an dua ve dang huu ti Nhan xet: Truac dat an phu, mot so trucmg hop chung ta cdn nhom cdc so hang phu hap, hoac chia cd hai ve cua phuong trinh cho mot dai luomg phii hap Vi du Giai phuong trinh yJx + \ V - x + \j4 + 3x-x^ =5 L&igidL *) D i k kien: -1 < x < Dat VxTT + V - x = / 101 Suy V + J C - X ^ = Khi phucmg trinh da cho trd t + =5 o Tir giai dirge nghiem x = 0, x = Vi du Giai phuong trinh (x^ +1)^v2 = xVx^ +2 + Lcfigiau Dat x4x^ cho trd +2 = r Suy (x^ +1)^ = +1 Khi phuong trinh da - ? - = 0<^ t = -\ +) Voi ? = -1 ta CO xVx^ +2 = -1«< x < « X = -V-I + V2 x'*+2x^-1 = +) Voi ? = 2, giai tuong ty trudng hop tren ta dugc x = V-T+Vf Vay, phuong trinh da cho c6 nghiem x = +V2 , x =^ + V s Vi du Giai phuong trinh SxVT- x^ = - f x + V l - x M £of^lai *)Di§ukien:-l<x<l Dat X + Vl-x^ = ? Suy x^\-x^ = ^ Khi phuong trinh da cho trd +) Voi ? = - ta CO X = i - * ! ^ +) Voi t = — , khong c6 nghiem Vay, phuong trinh da cho c6 nghiem x = \±4i 102 Vidu4 6x + ylx + 2+2yf3^ = Syl6 + x-x^ L&i gidL *) DiSu kien: -2 < x < ' Dat VA: + + V - x = / > S u y r a A^6 + x-x^ - x = f^-14 f Khi phuomg trinli da clio tra r = - ) <r> / = Taco Vx + + V ^ = 4<»4V6 + x - x ^ -3x = Tir giai nghiem x = Vi du Giai phuong trinh xVx + + x + = ,^ LM gidL *) DiSu kien: x > - Dat Vx + = ^ > Khi phuong trinh trd 't = \ P+t^ -3r + l = 0<» _/ = - l + V2 ' Tu giai nghiem x = -2, x = -ly/l * Vi du Giai phuomg trinh x^ + \/x'* - x ^ = 2x +1 " LdigidL Nhan thay x =^ khong phai la nghiem cua phuomg trinh da cho Chia ca hai ve cua phuong trinh cho x ta duac x - — + l x - — = X V X Bat ^x- — =t phuomg trinh tren tra P +t = 2<^(t-\)[t^+t + 2) = 0ot = \ Tir ta CO nghiem ciia phuomg trinh ban dSu x = itVs Vi du Giai bdt phuomg trinh > x - V x + x + Vx + LdigidL *) Qihu kien: x > Nhan thdy x = khong thoa man Chia ca tu va mau cua cac phan s6 a \& trai cho Vx ta dugc Dat V ] c + ^ = / > tadugc +1 >2 ^ Tir giai nghiem cua phuong trinh x = \ Vi du Giai b i t phuong trinh 2x + 5^ > 11 + 14 jc-2' LMgiaL Dieu Icien: < x 9^ Bat phuong trinh da cho tro ( x - ) + 5V]^>7 + ^ ^ <^2{x-2) + 54^> x-2 Ix x-2 Ro rang x = khong thoa man bat phuong trinh (1) Voi < X ^ bat phuong trinh (1) tuong duong voi 4~x x-2 x— Dat — = - = t K h i bat phuong trinh tro 2/ + > y « ^ ^ - ^ y ^ > c^t{2t + 7){t-l)>0 Xet dau ve trai cua (2) ta c6 't>\ (2)o — <t<0 ^^r-1 ' , u ( V ^ + l)(>/^-2) ^ +) V o l / > ta CO —j=^ > 1, hay -^^^^ > <=> x 4~x 4~x x-2 +) V o i — </<0 taco — < —1=^<0, hay 2 yjx (Vx+4)(2Vx-l)>0 0<x<2 104 Vay bat phuong trinh da cho c6 nghiem la x>4,—<x<2 V i du Giai b i t phuong trinh yfx + yj\-x^ > V2^3x^^4x^ x>0 L&igidL Dieu kien: l-x^>0 ^0<x< -3 + V41 .2 - X - J C ' >0 Bat phuong trinh da cho tuong duong voi +2ylx{l-x^)>2-3x-4x^ x + l-x^ 3(x' + x) - (1 - x) + lyjix + ^ X^ + X <»3 1-x x^ +x ^ + 2, )(1 - x) > -1>0 V 1x> „^^^>-«9x'+10x-l>0<» \x x< -5 + V34 -5-N/34 Ket hop dieu kien (*), ta suy nghiem cua bat phuong trinh la -5 + V34 -3 + V4I <x< V i du 10 Giai b§t phuong trinh x^ > ( l - V x ) ( x - V x + 3) Z,o"/^/ai Dieu kien: x > Nhan thiy, moi x > la nghiem cua b i t phuong trinh Xet < X < K h i bat phuong trinh tuong duong voi >2x + ( l - V ^ ) l-^x >2 1-V^ + 3<^ 1-V^ X >3 <-l Suy nghiem ciia b i t phuomg trinh la — — < x < \ +) Vori — ^ < -1 <::> X - Vx +1 < 0, v6 nghiem l-y/x Vay nghiem cua bdt phuomg trinh la x > V i du 11 Giai b i t phuomg trinh L&igidL *) Dieukien: ' f ' r- ' 2(x^ + 2)> V ? + l x-'+l>0<=>x>-l Bat phuomg trinh da cho tuong duomg v o i V, 2(x^ - X +1) + 2(x +1) > 5^(x + l ) ( x - x + l ) x^-x +1 c • 'J \ rx :r o +2 - ^ > U ± i - Dat Vx^-x+1 T X + x ^ - x + = r > t a d u g c 2r -5t + +) V o i ^ < - ta CO nghiem - < x < ^ ^ 2 2>0^ ^>2 ;x > tt^!^ 1/40 St v,b JV - + ) V o i f > ta CO 4x^ - 5x + < 0, v6 nghiem Vay bat phuong trinh da cho c6 nghiem - < x < - — ; x > ^ ' ^ ^ ^ 2 V i du 12 Giai b i t phuomg trinh ( x + - V2x + )^ + 3V2xTT > 2x + Lố^iflị * ) D i e u kien: X > - — ' ' :r Dat V2x + = r > Bdt phuomg trinh da cho t r d (/-l)"* -3r+3 <^(?-2)r?(/-l)2 >0 +1 106 or>2,vi tit-iy+\>i,yt>o Suy nghiem ciia bat phuang trinh da cho ^ - • .1 Vi du 13 Giai hk phuomg trinh 2x^+x + Vx^ + + 2x^x^ Lei giai Bat x + yl,x^+3=t Taco 2x^ +3 + 2xylx^ Chuy r i n g + < t>0 +3^t^ Khi bat phuang trinh da cho tra t^+t-\2<0<^(t + 4)it-3)<0=>t<3 Suyra x + \lx^ +3 < ^yjx^ +3 <3-x<^x<l Vi du 14 Giai bat phuang trinh yjx-yjx^ - \ ^x + ylx^- \ ^ x^-l>0 L&igidL *) Di8u kien: x-yjx^ - >Oc:>x>l :+ Dat >0 - = f > K h i bat phuang trinh da cho tra iV s y + / < | <^it-2){2t^ +4t-\)<0 Suyra ^x + yjx^ - I <2^x + b) Ddt dn dua ve he phuomg trinh ot<2,\\2t^+4t-\>0yt>\ ylx^-\<\6<^\<x<^ 32 / Nhan xet: Phuomg phdp ddt dn phu hodn todn de dua ve he phuomg trinh thumg duffc dp dung vdo giai cdc phuang trinh dang u" {x) + b = a^au{x)-b hodc cdc phuang trinh chica hai cdn thicc; moi can thicc duac ddt cho mot dn phu, sau khai thdc quan he giita hai bieu thicc cdn de thiet lap phuang trinh cho hai dn moi 107 Vi du Giai phuong trinh yjl-x + ^l + x = L&igidL Dat ^l-x = u,^l + x = v ta thu dugc he w+v=2 w + v = <=>M = M+ V = (M + v)^ - 3MV(M + v) = [wv = l V Tu giai nghiem cua phuong trinh x = Vi du Giai phuong trinh V x - + yjl-x = L&igidL *) Dilu kien: X > Dat yjx-l = w > , \ / - x = V ta dugc he phuong trinh "v = w + v =1 •(l-v)2+v^ = l v=l v = -2 Tu giai dugc nghiem cua phuong trinh da cho x = 2,x = l,x = 10 Vi du Giai phuong trinh V - x + VT+Sx = L&igiai *) Dieu Icien: < x < Dat V - X = w > 0,^1+ 5x = v > ta thu dugc he 'M + V = ^,=^5(3-v)2+v^=21 =21 v=l «(v - l)(v - 2)(v^ + 3v +12) = o v=2 Tir giai dugc nghiem cua phuong trinh da cho x = 0, x = Vi du Giai phuong trinh V4-3Vl0^^3x = x - 10 x< 4-3VlO-3x>0 L&igidL *) Di6u kien: Dat x - = ;; ta dugc - ^ - = 108 Tigp tuc dat yl4-3y = z > ta thu ducrc he • y=4-3z z^=4-3y Giai he phuong trinh tren thu dugc y- z = l Tu thu dugc nghiem cua phuong trinh da cho x = Vi du Giai phuong trinh V2 + x + yl3-x = + ^(2 + jc)(3-x) L&igidL *) Biku kien: -2 < x < Dat yl2 + x =u> 0,V3-x = v > ta thu dugc he V= u +v =3 uv = Tir giai w, v roi sau giai dugc nghiem ciia phuong trinh da cho X = -\,x = Vi du Giai phuong trinh yj3-x +1 = x^ - x W + + wv L&igidL *) DiSu kien: x < Dat x-\ y phuong trinh da cho tro tharih 72^ +2= / Tigp tuc dat yj2-y = z > ta thu dugc he / =z + >; + {y + z)(y-z) Tir giai dugc z = 2, z = =y +z -i+Vs Tu giai dugc nghiem cua phuong trinh x = - l,x = •I- X + 75 z Vi du Giai phuong trinh 6\/6x + + = x'' L&igidL Dat yf6x + = y ta thu dugc he x^=6>' + , z^(x-y){x +xy + y +6)^0ox = y /=6x +4 Tu giai dugc nghiem cua phuong trinh x = -2,x = ± 73 -V- X C V- V i d u Giai phuongtrinh ^{3x + lf Lai giai Dat + = a, ^3x- + ^{3x-lf =l (a-b)[a^+b^+ab) =\ + ^9x^-1 =1 \ ta c6 a^+b^+ab a'-b'^2 a^+b^+ab a-b (Jb + lf +b^ +{b + =2 l)b^\ =2 Tir giai nghiem cua phuomg trinh da cho la x = Vi du Giai phuong trinh 4x^ -13x + V3x +1 + = ' ~ ^ ' ''^ ^-i^im m^mv r „ f._ j _ Lo-Z^/fl/ *) Dieu kien: X > - - Nhan xet: Khi dat an phu de dua ve he phuong trinh chung ta thuang dua ve he doi xirng hoac he phuomg trinh ma sau cong dgi so hat ve cua no ta dua ve duac phuang trinh tich Muon vdy mot so truang hap ta can chu y den "he so" cuaphep dat CJ vi du ta dat VSxTT = ay + b roi chon a,b de c6 dugc he nhu mong mu6n Dat V3x +1 = - 2j; ta thu dugc he ( - x f = x + 2>; + l (x->;)(2x + 2>;-5) = (3-2>;) - x + l Tu giai dugc nghiem cua phuang trinh ban dau 15-V97 x= 8: 11 + V73 ,x = Vi du 10 Giai bdt phuang trinh Vx + + ^ - x > L&igidL *) Dieu kien: x > - Dat Vx + =a> , ^ - x =b la thu dugc he 110 a + b>l \ =>a^+ <3 a'+b'=3 a > l + V2 x > + V2 -2<x<2 0<a<2 - X>g + ( C - X]- Da/ an (^ira ve phuang trinh tich Vi du Giai phucmg trinh 5VJC^TT = 2{x^ +2) L&i gidi *) Di^u kien: x > - * Dat Vx+T = w>0,Vx^ + = V > taCO + = + Khi phuong trinh da cho tra l£ w 5WV = ( M ^ + V ^ ) O ( W - V ) ( M - V ) - Tir giai dugc nghiem cua phucmg trinh da cho x = + V37 Vi du Giai phucmg trinh x^ + 3^x^-1 = Vx'* -x^ +1 :i L&i giai *) Dilu kien: x < - l;x > iiiiO Dat x^ = a > 1, Vx^ - = > ta dugc phucmg trinh <^6(3a + 56) = «6-0, vi 3a + 56>0 a + 3b^yla^-b^ Tir day thu dugc nghiem cua phucmg trinh ban dku x = ±1 Vi du Giai phucmg trinh %/x + l + N/X^ = y[x + ylx^ +x L&i gidi Dat N/X + =a, ^ = b ta dugc phucmg trinh [a^b a + b^ =^b + abc^(a-b)ib-\)^0 = <:> X = Vi du Giai phucmg trinh Vsx^ +14x + - Vx^ - x - = 5VxTT fx>-l L&i gidi *) DiSu kien: <i 5x^ + 14x + > o x > Phuong trinh da cho t u o n g duong v o i ' ^ yJSx^ + 14x + = Vx^ - X - 20 + « x ^ +14JC + = X ^ - x - + 25(x + l ) + 10V(x + 4)(x^ - x - ) < » ( X ^ - J C - ) + 3(X + ) = V ( X + ) ( X - X - ) Dat VA: + = a, yjx^ -4x-5 26^ + +) V o i a-bta =b ta dugc phuong trinh = 5a6 <=> (a - 6)(3a -2b) dugc nghiem x = ^ =0 • + ) V o i 3a - 26 = ta dugc nghiem x = Vay phuong trinh da cho c6 nghiem x = ^ V i du Giai phuong trinh 3xylx^ +1 = x V , x = -19x -16 L&i gidi *) D i e u kien: x^ + l > < » x > - l Phuong trinh da cho t u o n g duong v o i 3xyl(x + l)(x^ - X +1) = ( x ' +1) + (x^ - X +1) - ( x +1) Dat a = b = y[7 -x + \,a>0,b>0 K h i phucmg trinh t r o 3(a'-l)aZ) = a V + ' - o ' <=> a^b(3a -b) = b(3a -b) + 2{b^ - 9a^) ^i3a-b){a-b + b + 6a) = <;=>3a-b = 0, v\ + b + 6a>0 Suy 3Vx + l = Vx^ - x + « x ^ - x - = < » x = ± V 3 , t h o a m a n d i g u k i e n Vay nghiem cua phuong trinh la x = ± V33 V i du Giai phuong trinh x + ^ - x = V x - l + V-x^ + x - + L&i gidi * ) D i e u kien: l < x < 112 Dat yJx-\ a, yjl-x -b ta dugc phuomg trinh + 2b = 2a + aboia-2)(a-b) = +) Vai a = 2, dugc nghiem x = +) Vai a = b, dugc nghiem x = Vay nghiem cua phuang trinh la x = 4;x = V i du Giai phuang trinh V4x2+5x + l - V x - x + l = x - L&i giai Dat V4x^ +5x + l = a, 2\lx^-x trathanh a-b^a^ -b^ <^{a-b){a + l = K h i phuang trinh da cho + b-\) = Q +) Vai a = b thu dugc nghiem ^ = + ) V a i a + b = \o \ / x ^ + x + l + V x ^ - x + = (1) Chu y ring ve trai cua (1) luon Ion han vai mgi x , nen (1) v6 nghiem Vay phuang trinh da cho c6 nghiem nhdt ^ ^• V i du Giai b i t phuang trinh 2(1 - x ) V x ^ + x - l < x^ - 2x - L&i giai *) DiSu kien: x^ + 2x - > 0, hay x < - - V ; x > - + V2 Dat yjx^ +2x-l = (3 > 0,1 - X = 6, b i t phuang trinh da cho tra 2ab<a^-4(\-b)^(a-2)(a o Vx' + x - l - + 2-2b)>0 V x ^ + x - l + x >0 (1) Xet truong hgp sau: TH1 x>-\ V2 Khi Vx^ + x - l + x > , nen (1) tuang duang vai x'+2x-l-2>0 x > - l + V6 • x > - l + V6 x<-l-V6 77/2 x < - l - V Khi V x ^ + x - l + x < , nen (1) tra > / x ^ + x - l - < < » - l - V < x < - l + V6 113 Ket hop dieu kien cua trucmg hop ta dugc - - V < x < - - V2 Vay nghiem cua b i t phucmg trinh da cho la - - V6 < x < - - V2 ; X > - \ y/6 Vi du Giai bat phucmg trinh x - > x^^x-l Dat yfx - a, y/x-l b^>a^ib~a) Chu y rang b-a<0, - Vx j + \Jx^ -x = 6, bat phucmg trinh tren trd + ab<:>ia^-b){b-a)<0 nen bat phucmg trinh tren tucmg ducmg vai >b, hay x>y/x-\ Bat phucmg trinh nghiem dung vai moi x>l Vay, bat phucmg trinh da cho c6 nghiem x > ' Vi du 10 Giai b i t phuang trinh {x +1)^ < ( + ^x^ -2x^ - 3x + x> LM gidu *) DiSu kien: x^ - 2x^ - 3x + > < X < \ Bat phuang trinh da cho tra (x^ - X- 4) + (x -1) < ^ ( x -1) (x^ - X- 4) I + V17 Truang hap x> at ylx-\ a>oJx^ Dat -x-4=b> ta duac b^+3a^ <4ab<:>ia-b){3a-b)<0 TIT day ta dugc - <a<b, >/x^<\/x^-x-4 hay 3yf7-i>ylx^-x-4 Tir giai dugc nghiem < x < + 2V5 • ' 114 Truomg hap — ^ — < x < Dat V l - x = a > , V - x ^ + x + = > ta duac -b^-3a^ < Aab, luon thoa man Vay nghiem cua hk phuomg trinh da cho < x < + iVs ; -—'^^^ < x < 'I ? Bai tap phan 1.4 Bai Giai phuang trinh 4x^ - 3x - = ^x^ -x^ Bai Giai phuomg trinh V3x-2 + Vx-1 + = 4x + 1^3x^ -5x + l Bai Giai phuang trinh V2x + + Vx + +16 = 3x + l^lx^ +5x + Bai4 Giai phuang trinh 2x^+(x-l)Vx+T = 5x +7 _ Bai Giai phuomg trinh 3^2+ x - ^ - x + A^A-x^ = 10 - 3x (DkthiDHB 2011) o , Bai Giai phuang trinh x^ + 2x X — = 3x + l Bai Giai phuang trinh 2x^ + 5x - =X7^x^-1 Bai Giai phuomg trinh 4x^ + (4x - 3)>/x-l = 5(x +1) Bai Giai phuang trinh Vs + x + = x^ - 4x Bai 10 Giai phuang trinh >/l5 + 2x + 20 = 32x(x + 1) j .T r ^miaiv:- - • IX + Bai 11 Giai phuomg trinh 2x^ + 4x = J—^ Bai 12 Giai phuang trinh 3^8 I x - = 3x^ - 6x^ + 4x - Bai 13 Giai phuomg trinh ^5 + x + ^ - x +1 = \/x^ +3x-10 Bai 14 Giai phuang trinh Vx+T > + %/x Bai 15 Giai phuang trinh (3x - 6)Vx'-l = x' + x' -17x +18 115 Bai 16 Giai phucmg trinh yjx + l + ylx + = + \lx'^ +3x + Bai 17 Giai phucmg trinh 7(x^ + x) = 4x + 28 Huong dan bai tap phan 1.4 R6 rang x = khong thoa man phuong trinh da cho Chia hai X - cho x ta dugc Dat | x - i = / tadugc 4/^-?-3 = 0<=>(r-l)(4r^+4/ + 3) = <^ r = T u giai dugc nghiem cua phuong trinh ban dku x = + Vs Digukien: x > l Dat V x - + V x ^ = / > Binh phuong hai cua (1) ta dugc 4x + 2V3x^ - 5x + Khi phuong trinh da cho tro t'^ -t-6 = 0<^ (1) 2=r+3 t=3 t = -2 Suy V x - + V ^ - = Giai phuong trinh dugc nghiem x = Tuong tu bai DiSu kien: x > - Dat Vx + =t>Q Khi phuong trinh da cho tro f ( / ^ + ? - / - ) = Phuong trinh truong duong voi t{t - 2) {2t^ + 5^ +1 j = «> / =0 /=2 Suy nghiem cua phucmg trinh ban d i u x = - l , x = Di^u kien: - < x < Dat V2 + x - ^ - x =t {\) Binh phuong hai v6 cua (1) ta dugc 10 - 3x - A^JA-X^ = Khi phucmg trinh da cho tro 3/ = o f=0 r = 3" 116 ' Thay vao (1) giai dugc nghiem cua phuong trinh ban dau ^ ~ ^• -l<x<0 Dieu kien: x — > x>l X Chia ca hai ve ciia phuong trinh da cho cho x ta duoc x — + X Dat ^ X = / > ta duoc + 2t-3 = X 0c^t^\ Tu giai dugc nghiem cua phuong trinh ban dau x = 1±V5 Di6ukien: x > l Phuong trinh da cho tuong duong voi 2(x^+x + l ) + ( x - l ) = ^ ( x - l ) ( x ^ + x + l ' « + - ^ = - = X X + X+ VX + X+ >-2 x-1 Oat ^ +X + x-1 = / > Khi ta CO - 7/ + = <=> 1 • t= +) Voi / = ta dugc 4x + 3x + = 0, v6 nghiem +) Voi f = ^ ta dugc nghiem x = ± V6 Dieu kien: x > Dat 2x + ylx-l = t Khi phuong trinh da cho tro t = -\ r-3/-4 = « t = +) Voi ^ = - , khong thoa man +) Voi t = ta CO nghiem ciia phuong trinh ban dau x = 117 Dieu kien: x > - Dat x-2 = y tadurgc + +7 = Dat ^y + = z > ta dugc he / = z +7 (y-z){y + z + \) = z^^y + +) Vai y = z giai dugc nghiem x = + V29 +) Vai y + z + l-0 giai dugc nghiem x = - 10 Dieu kien: x > - — Phuang trinh da cho tra Vl5 + 2x + 28 = 2(4x + if Dat 4x + = a,yj2x + \5 = Z? > ta dugc 2^2 =a =6 + 28 2b^ 2S <^{a-b){2a + 2b + l) = Tu giai nghiem cua phuang trinh x = ~,x = ——^^^^ 16 11 Dieu kien: x > - Phuang trinh da cho tuong duang vai 2(x +1)^ =,/ "'"^ + V D$t x + l = a, x + -b>0 ta dugc he 2a'=b + 2b^ =a + Tir giai dugc nghiem cua phuang trinh ban dhu -s-VTs -3 + 4vi x,x = <4 12 Phuang trinh da cho tuang duang vai 118 27^27(3x - 2) + 46 = (3x - 2)^ - 46 Dat 3x - = a, ^27a + 46 = Z) ta dugc 27b = a'-46 27a = 6^-46 ia-b)i^a^ +ab + b^ +n) = 0<^a Khidotaco a ^ - a - = 0<»(a + ) ( f l - a - ) = 3+ 2V6 Tir nghiem=cua 13 do^5 thu + xdugc = a,^2-x b phuoTig tadugc trinh da cho x = 0,xa + b + ab = -\ a + b + ab = -l {a + bf -3abia + b) = l 'a = ~l,b^2 a = 2,b = -\ Tu thu dugc nghiem cua phuong trinh da cho x = -6,x-?> 14 Dieu kien: x > - Dat 4x + \ a , ^ = taCO b<a-\ , / ,\ a - 11 = 0A3 ( a > ) ^ a ^ - l < ( f l - l ) 0<a<l a>3 -l<x<0 Tu thu dugc nghiem cua bit phuong trinh ban diu x>% ' 15 Dieu kien: x>l.Phuong trinh tuong duong vai (3x - 6)V(x-l)(x^+x + l) = x(x^ + X +1) -18(x -1) Dat a = yfx^, b = Vx^+x + 1, a>0,b>0 Khi phuong trinh tra i3a^ -3)ab = (a^ +\)b^ -\Sa^ ^3a^b-3ab-a^b^-b^+\Sa^ =0 « {3a'b -a'b^)-(3ab-b') +(\Sa' -2b^) = o a^bi3a -b)- b{3a -b) + 2{3a - b){3a + b) = <^(3a-b){a^b + b + 6a) = 0c>3a-b=-0 (vi a>0,b>0) Suy sVx-l = \lx^ +x + l ^9(x-\) = x^ + x + \ox^ -Sx + \0 = 0^x = 4±y[6 Vay nghiem cua phuong trinh la x = ± V6 16 Dat = a,^[7+2 = ta dugc a + Z) = l + afe<=>(a-l)(6-l) = ^ a = l x = =1 x = -l 4x + 17 Dat x + - = >;, J = -y + — = z thuduoche V 4 Tir giai nghiem ciia phuong trinh ban dSu x = ^'^^^^ ^x= ^ 28 V7 28 • • 14 14 1.5 Phuangphdp ddt an phu khong hodn todn a) Dat an dua ve phuong trinh tham so la an ban ddu Vi du Giai phuong trinh 2(1 - x)Vx^ + x - l = - 2x - L&igidL *) Dieu kien: x^ + 2x - > Dat V ? + 2x - = / > Khi phuong trinh da cho tro / ^ + ( x - l ) / - x = (1) Xem (1) la phuong trinh bac hai an t (x la tham so) ta giai ducrc nghiem 't = J = -2x +) Voi ^ = ta CO Vx^ + x - l = x = -1 ± V6 +) Voi t = -2x ta CO -y/x^ + x - l = - x , phuong trinh v6 nghiem 120 Vay, phuang trinh da cho c6 nghiem x = - ± >/6 Vi du Giai phuang trinh (4x - \)\lx^ + \ + 2x +1 L&igiaL Dat Vx^ +1 = ^ > Khi phuang trinh da cho tra ^^+2(x-l)/-4x = « ( ^ - l ) ( ? - x + l)= < » / = x - l , vi r - l > Tu giai nghiem cua phuang trinh da cho ^ = - • x2+2 x = l +2 Vi du Giai phuang trinh x^ + Lotigidi Dat Vx^ + = r ta thu dugc phuang trinh r - (2 + xV + 3x - = < ^ _/=x-r Tu giai dugc nghiem cua phuang trinh ban dSu x = ±V7 Vi du Giai phuang trinh (x + \)^x^ - x + = x^ + LMgiaL Dat - x + = / > K h i phuang trinh da cho tra ^2+(x + i y + 2(x + l ) = o ( t - ) ( / - x + l ) = +) Vai ^ = ta c6 Vx^ - x + - x = ± V2 +) Vai ^ = X - ta CO Vx^ - x + = x - ,phuang trinh Vhong c6 nghiem Vay, phuang trinh da cho c6 nghiem x = ± V2 Vi du Giai phuang trinh yj%{x + 2) + V - x = VQX^ +16 LMgiai *) Dieu Icien: - < x < Binh phuang hai ve ciia phuang trinh da cho ta dugc x + x - = 16V8-2x^ (1) Dat \ / - x ^ = ? > Khi phuang trinh da cho tra +16r - x^ - 8x = {It - x){lt + X + 8) = 121 +) V o i 2t-x = giai dugc nghiem x = +) V a i 2/ + X + ta c6 2y]^-2x^ 4V2 +x + S = 0, phuong trinh v6 nghiem 4>/2 Vay, phuong trinh da cho c6 nghiem x = V i du Tim nghiem duong cua phuong trinh II - — + 3., \ \ L- — = ^^Ljt^l X l-i>0 LMgidL *) Dieu kien: X x >0 X Vi X> suy X> Phuong trinh da cho dugc viet lai duoi dang x-1 x-\ Dat t^-{l x-1 Khi ta dugc phuong trinh -t>0 + 3^Ix + \)t + 2x = Giai phuong trinh tren nhu phuong trinh bac hai theo t ta dugc f =2(l +V ^ ) ; ^ = - l + V ^ +) V o i r = 2(1 + V-v + ) ta CO nghiem v i x-1 =2^1 + Vx + l ) , phuong trinh v6 < l < f l +Vx+TVVx>0 x-1 +) V o i t^-\ Vx + ta CO V = - + v x + Binh phuong hai ve thu dugc ( x - V x + T = 122 Tu day giai nghiem cua phucmg trinh ban dku x - Chuy: Co the dat ^ = a, yJx + \ b dk dugc phuong trinh {a-b-l){2a-b-2) =0 b) Dat an phu dua ve he chita an ban ddu Vi du Giai phuong trinh ^2~x +2 = x^ Led giai *) Digu kien: x < Dat V - X = y > Khi ta dugc he x^ =y + {x-y){x + y)^x +y +) Voi x + y = Q taco x + V - x = 0<::>jc = - +) Vai X- = ta CO X - V - x = <::> x = i - t ^ Vay, phuong trinh da cho c6 nghiem x = ,x = -2 Vi du Giai phuong trinh 4x^ + 7x +1 = 2^x + L&igidL *) Di^u kien: x > - Dat 2x +1 = r, V / - x = y thu dugc he r + x = 2v _^(t-y)(t y^+3x = 2t +) Voi t- y-O + y + 2) = ta CO 2x +1 = Vx + Giai nghiem x - — +) Voi t + y + = giai duoc nghiem x = — Vay, phuong trinh da cho c6 nghiem x = - x = — 4' 10Q Chuy: Ta cung c6 ihk dat Vx + = / > d§ duoc phirong trinh (/ + !)' ( / - ) ( / - l ) = Vi du Giai phuong trinh 8x^ - 13JC^ + 7x = N / X ^ + X - L&igidL Viet lai phuong trinh da cho duoi dang {2x-Xf -(x^ - X - ) = ^ ( x - l ) + x - x - l Dat x - l = a,>/x^ + x - =fe,tir(l)tathudugc - x^ + x + l = 2Z) (a-6)(a2+a6 + 62+2) = ^ - x ^ + x + l = 2a a = 6, vi +fl6+ 6^ + > T u d o t a c o N / X + X - = x - l <»(x-l)(8x2-5x-2) = x=l 5±V89 X = 16 Vi du Giai phuong trinh x^ - 6%/6x + = Ldtigial Dat A/6X + = ^ ta dugc he phuong trinh x-' - 6>' = I J \ y - x =4 ^(x- y)(x +xy + y +6] = ^ ^ <=> X = J', vi x^ + x^^ + + > Tir ta CO x = >/6x + <:>x^-6x-4 = 0<=> x = - = 1±N/3 X Vi du Giai phuong trinh 2x - x - = v6x + L&igiai *) Dieu tcien: x > - ^ Dat V6x + = 2>' > ta thu dugc he phuong trinh 124 2x^ =x + \ 2y ^(x-y)ix + y) = -(x-y) 2y^=3x + \ x= y x + y = -\' Tir giai dugc nghiem cua phuong trinh da cho x = 3±vr7 Bai tap phan 1.5 Bai Giai phuong trinh x^ + 2{x - l)Vx^ + x + l = x - Bai Giai phuong trinh x^ - x + = (x + l)Vx^ - x + Bai Giai phuong trinh 4yjl + x = + 3x + 2y/\-x + V l - x ^ X + + xV2x + l n>- ^ r - u Bai Giai phuong trinh v2x + l I T" — - y j x +2 Bai Giai phuong trinh 3x^ x ++ 2x + = (3x + l)yjx^ +3 H u a n g dan bai tap phan 1.5 Dat Vx^ + X + = t, phuong trinh da cho t r d r ^ + ( x - i y - x + l = 0<»(/-l)(/ + x - l ) = .1 nb Dat V ? - 4x + = f > K h i phuong t r i n h da cho t r a r - (x + > + X= <=> t=\ t = x' T u giai nghiem cua phuong trinh da cho x = , x = - Dieu kien: - < x < Dat V l - x = / Phuong trinh ban dau t r a - (2 + VrT7)/-2(1 + x) + V m - t = 2yl\ ^ t = 2-sf\ ^' 125 Tir giai nghiem ciia phuong trinh da cho x~0,x Dieu kien: = ^X + yj2x + \^0 Dat y/x + -t ta duoc phuong trinh - { ^ X + y/2x + l'jt + Xy/2x + \0 <^ (t-x)(^t-yj2x 't^x + 1^ = J = yj2x + \ T u giai dugc nghiem ciia phuong trinh da cho x = \;x = Chu y: Cung c6 thk dat yjx + =a>0, yj2x + \>0 {a-b)[a^ -a-2) d% dugc phuong trinh = Dat \Jx^ +3 = t, phuong trinh da cho tro -{3x + l)t + 2xix + l) = 0c^{t-2x){t-x-\) = Tir giai nghiem cua phuong trinh da cho la jc = 1.6 Phirmgphdp ddnh gid Vi du Giai phuong trinh yJ2-x^ + x+ - V Lei giai *) Dieu kien: ^ - - 2• Phuong trinh da cho tuong duong voi 42-x^ ^ 1 +x + - — + - =4 x^ X (1) Theo bat ding thuc Bunhiacopski ta c6 4r- + X< va X Suy 42-x^ X +x+ X^ X 126 Do phuofng trinh (1) tucmg duong voi V ^ 1 ^ 2-—+= Vay, phuong trinh da cho c6 nghiem nhit x = V i du Giai phuong trinh Vx + + yjlx-l + V13-3JC 13 ^/ai *) Dieu kien: < X < — Theo b i t dSng thuc Cosi ta c6 V ^ + A/2X-2+V13-3X = ^ ( V ^ ) + ^ ( V ^ ) + ^ ( V l - 3x) 4+ x+ l + 2X-2 + 13-3X 4 , Dau dang thuc xay x = Vay, phuong trinh da cho c6 nghiem nhat x = V i du Giai phuong trinh x + V T - x ^ = -yjs-x^ L&igidL *) Dieu kien: |x| < Tu dieu kien cua x ta c6 x + yjl-x^ >-!>!- Vs-x^ Dau dang thuc xay x = -1 Vay, phuong trinh da cho c6 nghiem nhat x = - Vidu4 V ^ - V x - x + l-x^+3x-2 = L&i giai *) Dieu kien: x > Phuong trinh da cho tuong duong voi Vai X > ta c6 -(x-ir(x+i)<o Dau dang thurc xay x = Vay, phuang trinh da cho c6 nghiem nhdt x = V i du Giai phuang trinh N / - X + Vx + = ^ (x -1)^ L&igidL *) Digu kien: -1 < x < Ta CO ( V - X + VxTT)^ = + V - x V x + T > Suy +1 > 2, dau dang thurc xay x = -1 hoac x = (1) Mat Idiac, t u dieu kien cua phuang trinh ta c6 -2 < x - < 2, nen ^ ( x -1)^ < 2, dau dang thurc xay x = -1 hoac x = (2) Tir (1) va (2) suy nghiem cua phuang trinh da cho la x = -1 hoac x = Vi du Giai phuang trinh \l2x^ - + x V x - l = 2x^ L&igidL *) Dieu kien: x > Phuang trinh da cho tuomg duang vai 72x^-1 V2x-1 = , V X - , , V2x^-1 <1 Vai dieu kien cua bai toan ta c6 <1 va X x' Suy — z —+X < X Dau dang thurc xay x = Vay, phuang trinh da cho c6 nghiem nhit x = Vi du Giai phuang trinh L&igidL + x - +V4-x2 +yl2x-x^ =3 *) Dieu kien: < x < 128 T a c o ||jc| + V - x ^ J = + 2|x|V4-x^ > Suy |x| + V - x ^ > (1) • •" "• Dat |x - | = /,0 < / < 1, v i < X < Khido x-l\ hx-x^ ^t + Ket hop vdfi (1) ta dirac + ^fl^ : - l + V - x +^l2x-x^ D a u d a n g t l i u c x a y r a k h i x = hoac >3 x-2 Vay, phuang trinh da cho c6 nghiem x = hoac x = Vi du Giai phuang trinh Vx + + Vx^ + x + = 2x + V x + T L&i giai *) DiSu kien: ^ ^ - ^ • :1 Phuang trinh da cho t u a n g duang v a i Vx + - V2x + = x - V ? + x + 1-x ( x - l ) ( x + 2) Vx + 2+>/2x + l 2x + \lx^+x +2 (1) C h u y rang v a i x > - — ta c6 2x + Vx^ + x + > Xet cac truang hap sau day X = 1, thoa man ( x - l ) ( x + 2) - - < x < l K h i d = >0> , nen (1) v6 Vx + + V x + l X + A/X^ + X + nghiem ( x - l ) ( x + 2) , , ^ < < —^ , , nen ( ) vo nghiem + V2x + 2x + V x + x + 1-x i x > l Khido V ^ Vay, phuang trinh da cho c6 nghiem nhat x = 129 Vi du Giai hk phuong trinh yflx + l > yjlix^ + 2) + x - L&igidL *) Dieu kien: x > - ^ Bit phuong trinh da cho tuong duong voi V2x + > N / X ^ + + X - Chu y rang, voi x > ta c6 * *"' yj2x + l < \ / x ^ + x + l = x + l = + x - l < yjlx'^ +4+x-l Daudangthucxayrakhi x-0 Vay, bit phuong trinh da cho c6 nghiem x = Vi du 10 Giai phuong trinh V x - ^ x - ^ x - = x^ L&igidL *) Dieu kien: x > - Phuong trinh da cho tuong duong voi N/2X-1 ^ X - X X _^ X Chu y rang, voi dieu kien cua bai toan ta c6 X X X Suy V2X-1 ^ x - ^6x-5 < X X X Dau dang thuc xay x = Vay, phuong trinh da cho c6 nghiem nhat x = Vi du 11 Giai phuong trinh ^2A:-1 + V x - = 2(V^ + VI Ldigiai *) Dieu kien: ^ - ^ • Phuong trinh da cho tuong duong voi V I - V x - l + V I - f e - l =2 130 Tnrac hettachi 2Vx - ^ I x ^ > (1) That vay, (1) tirong duomg voi — + >/ - — < , dung (theo bat dang thuc Cosi) X \ Ti6p theo ta chi ^ - ^2x-\ (2) Ta CO (2) tuong duong vai J - X \ That vay, theo bat dang thuc Cosi ta c6 V _ l , i , l \ X V X <2 Tir(l)va(2)tadugc ^ ^ - V x - + V ^ - ^ x - l > Dau dang thuc xay x = Vay, phuong trinh da cho c6 nghiem nhit x = Vi du 12 Giai phuang trinh IsVx-x^ + 9ylx + x^ =16 LcngidL *) Dieu kien: < x < ' Ap dung bat dang thuc Bunhiacopski va Cosi ta c6 u4x-x^ + W x + x^ < (13 + 27)(l3x- 13x^ + 3x + ?>x^) = x ( - x ) < ^ ? ^ ^ l ^ ^ = 162 Suyra u4x-x^ +9\/x + x^ < V x - x ^ = - V x + x^ <:>X = 5x = - x ^ Vay, phucmg trinh da cho c6 nghiem nhit ^ = ^ • Dau dang thuc xay Vi du 13 Giai phuang trinh V2x' - x - xV2x' + = "^^^ ~ ^ ^ ^ • X <0 LMgiaL *) DiSu kien: 2x^ - 2x > <=> x > l 131 Ap dung bat dang thuc Co-si ta c6 Suy yjlx"-2x -x^jlx^ +1 < - X = DSu dSng thuc xay <=> X = -1 -V2x = Vx' +1 Vay, phucrng trinh da cho c6 nghiem nhit x = - Vi du 14 Giai phuomg trinh xyllx^ +2 + V6x^ - =3x^+1 L&igidL *) Di^u Icien: 6x^ - > <=> ^ ^> 3' Ap dung bat dang thuc Co-si ta c6 x4^^^l2x.4^7^<^='-^; V^7r^=i.2.V^^<l±^=^ Suy xV2x' +2 + N/6X' - < 3x' + Dau dang thuc xay ^''^ = V6x'-2 <=> X = Vay, phuong trinh da cho c6 nghiem nhat x = Vi du 15 Giai phuong trinh xVx + + V - x = 2vx^hl L&igidi *) Dieu kien: - < x < • ] Ap dung b4t dang thuc Bunhiacopski ta c6 x V ^ + V ^ < ^(x' + l)(x + l + - x ) = 2Vx' +1 Dau dang thuc xay Vx + = x V - x <=> "x = x-l±V2' 132 Bai tap phan 1.6 Bai Giai phuang trinh V x - + V - x = x'^ -6x + \\ of> lri>l Bai Giai phuang trinh l^fx + ^\-2x = Bai Giai phuang trinh V3x + + yjlx + V l - x =3 Bai Giai phuang trinh =^ + -Jl-x 3+ Vl-x Bai Giai phuang trinh 3X(N/X-2 + V4-x) = +9 Bai Giai phuang trinh ^ x - + + ^ - x = x + Bai Giai phuang trinh x + x^ + yll-x"^ + ^2-x^ = Bai Giai phuang trinh 2x^ + V2x-1 = x^ + 2x Bai Giai phuang trinh x + 2yl2x-x^ = + 2Vx Bai 10 Giai phuang trinh VxTI + VxTs = 3x + V2x v v fu: Huang dan bai tap phan 1.6 Digukien: < x < V ^ + V4^<l±^ + i±^ = < ( x - f + = x - x + ll "r Dau dang thuc xay x = ^ ^: -v;:^ lyjid Dieu kien: < x < — ^• ^ i •ChuyrSng < l - x < l Suyra 2>/Gc + ^ l - x >2x + ( l - x ) = l Diu ding thuc xay x = 3.KhiD i k kien: ' ' ^ > , < ^x < .Suyra V3x + + V2x+>/l-x VV3x2 x+ +4v>r2^ > i Dau dang thuc xay x = 133 ,^ Diku kien: < x < Khi danh gia duoc 3+ <i<^ + yjl-x y/l-X Dau dang thuc xay x = \ Di^ukien: 2<x<4 Ap dung bat dang thuc Cosi ta c6 x ( ^ / ^ +^/4^)<3x l+x-2 1+4-x + Dau dang thuc xay x = Danh gia ^ x - < x va V ^ + ^ - x < Dau dang thuc xay x = y.Danhgia x^+yjl-x"^ <2 va x + ^2-x'* < Dau dang thuc xay x = Dieu kien: x > — Phuong trinh da cho tuong duong voi = (x -1)^ + X Taco ^—<1 < ( x - i f + l D a u dang thuc xay rakhi x = l X Digu kien: < x < Nhan thay x = khong thoa man Chia ca hai cho Vx ta dugc 2(V^ + > / - x ) = + V ^ + ^ ^ Vx Danh gia ( V x + > / - X ) < < + N/X+-^- Dau dang thuc xay x = 10 DiSukien: x > Phuong trinh da cho tuong duong voi > / x + T - V ^ - x - > / x + Chiatnronghop 0<jc< va x>l de so sanh hai ve voi Tir suy nghiem x = \ 2) Cac phirang phap lirgng giac va giai tich 2.1 Phuang phdp lugtnggidc hoa Nhan Xet: Khi gidi mot s6 bdi todn chica can thicc chung ta c6 the sic dung phep dat luang gidc (luang gidc hoa) de khu can thicc Khi dgt an phu luang gidc ta su dung phep dgt dgc trung theo dgc diem cua can thicc c6 bdi todn Cu the nhu sau: Gid sir a la so thuc duang +) Voi ^a-x^ , dat x = 4asmt,- — <x< — hoac x = Vacos/,0<x<^r 2 +)V6i V x - a , d a t x ^ ^ , t ' s'mt 0; TT• y ,te hoac x = cost hoac x = yfa cott,tE{0;7r) 2'2 +) Vori yjx^-a , dat x = Va tdnt,t Vi du Giai phuang trinh 4x^ -3x- \ll-x^ LcfigidL *) Digu kien: -1 < x < Dat X = cosf,0 <x<n Phuang trinh da cho tra '7t t^ 4cos^^ -?>cost-s,\nt <^ cos3r = cos v2 y n—+ kn r = 7>t^ t + kln — 3f = ^t^kln Do / e [0;;r] nen ta2 c6 cac gia tri cua ^ la = —,^2 = ~^'^3 ^ • 135 Suy nghiem cua phuomg trinh da cho x = cos—,x = cos—,x = - — 8 Vi du Giai phuong trinh + sVl-x^ = 8(x^ + (1 -x^)^) LMgiai *) Dieu Icien: -1 < x < Dat X = cos^O < X < Phuong trinh da cho tro + 3sinr = 8(cos^^ + sin^r) <=>5 + 3sin/= 8(l-3cos^/sin^^) <:> sinr = -8cos^rsin^ t ^ cos4f <=> cos4f = cos 2 TV —+ 10 Tt kin y2 J (ksZ) kin Do r G fO;;rl nen ta c6 cac gia tri cuat =t\k— t,+= —,U = —.U = — ^ ^ ' ^ ^ 10 Suy nghiem cua phuomg trinh da cho x = cos—,x = cos—,x = 10 10 Vi du Giai phuong trinh V2 (Sx^ -1) = 2x - Vl-4x^ L&i giaL *) Dieu kien: - — < x < — • 2 :; v ^ Dat x = -sin^,- —<^<— 2 Phuong trinh da cho tro V2 ^2 sin^ ^ ~ ^) ~sin/-COS/ V2 (sin/-cos/)(sin/ + cos/) = sinf-cos/ +) sin/-cos/ = 0.Khid6 t^ — ^kn,k^TL Do <t< — ,nen t = — Suy x = — ^ 2 'id 136 +) sin^-cos^>0 Khi V2(sin^ + cos?) = <=>sin t + — = — + k27r n 4J 12 (A:6Z) 7^ t + — = — + k27r / = — + )t2;r 12 Do < / < —, va sin/ - cosf > nen khong c6 t thoa man 2 <=> +) s i n f - c o s / < Khi V2(sin/ + cos?) = - <=>sin ? + —= + k2z , / = 5^ + k27r + A:2;7- r= f + —= 12 + A:2;T (keZ) Do - — < / < —4, va sin/-cosf <Onen t =12-— Suy x = - s i n — 2 12 12 —^, x = -sin 5;r Vay, phuong trinh da cho c6 nghiem x = >/2 12 35 Vi dy Giai phuong trinh x + L&igidL *)Di6ukien: x >\ Tu phuong trinh da cho ta thdy x>0 Suy x > Dat x = -^,te Khi phuong trinh da cho tra sin? = —35 •osmf + cos/ = 35 —sm/cosr smt cost 12 12 ; Dat sin/ + cos/ = M , l < M < V taco sin/cosf = Khidotaduoc ^ ^ ( u ^ - l ) 24 137 Giai phuong trinh tren ta dugc gia tri u thoa man la M = 1 - + - 35 Khi ta duoc • s'mt cost 12 1 25 sin? cos? 12 sin? sin? Tir ta dugc nghiem cua phuong trinh da cho la x = ^ , x = ^ 23 Vi du Giai phuong trinh Vl + x^ = — ^ ^ ^ ^ ) 6x^-20x^+6x LdigiaL *) DiSukien: -20x^ +6x<:>x ^Q,x^±yf3,x^±-7= v3 Khi phuong trinh da cho tuong duong voi 6x(l + x^)^-32x +x Dat X = tan ?, - y < ? < y , phuong trinh da cho tro cos? = 3sin2?-4sin^2? <»cos? = sin6? osin6?=:sin — ? v2 n k2n et = — -t + k2n: ? = — + 14 7t k2n: St = — + t + k27r t =— + 10 Do - — < ? < —, nen ta c6 cac gia tri cua ? la 2 ^ • _ 3;r _ n _57t _ 3;r _ n Tijr thu dugc nghiem cua phuong trinh da cho x = tan?,, / = 1,2, ,5 Vi du Giai phuong trinh ^ l + V l - x ^ U(}-xf ->/(l + x)^] = + V l - x ^ Lo'/^/fli Dieukien: - l < x < l 138 - : Dat x = cost,0<x<7r Phuong trinh da bho tro = + sin? yl\ sin? (7(1 -cosO^ - V(l + cos/)^ t t • ^ ^ sin—+ cos— 2V2 sin — c o s - = + sin? 2 2) it + sin—cos— = + sin? 4i sin• it— c o s — 2 o - V c o s / ( + sin/) = + sinf <^cos/ = — = « > x = ^ ^ V2 4i Vay, phuong trinh da cho c6 nghiem x = Vi du Giai phuong trinh 64x^ -Wlx^ + 56x^ - = 2yll-x^ (Di thi Olympic 30/4) Lei giai Dieu kien: - < x < Dat x = cost,Q<x<7t Phuong trinh da cho tro 64 cos^ r -112 cos"* t + 56 cos^ / - = \ / l - c o s ^ / (1) Nhan thSy cos / = khong thoa man (1) Do phuong trinh (1) tuong duong voi 64 cos^ r -112 cos^ t + 56 cos^ / - cos ? = cos r sin t <=> cos 7/ = sin 2? <=> cos 7? = cos — -It v2 It^ 2t + kl7r n k2n /=— + 18 n lt = -— + 2t + k27r k2n + 10 Do f e [O;^] va cosf ^ nen ta c6 cac gia tri cua t la _ n _57V _ I3n- _ \ln ^l8''2 "T¥''^ ^TF''^ = _'in _ln "To • Tu thu dugc nghiem ciia phuong trinh da cho x = sinfy,/ = 1,2, ,6 139 Vi dy Giai phuang trinh yl\-2x + -Jl + lx = x = -cost,0<t t <;r t l-2x l + 2x 1 Loti gidL *) Dieu kien: — < x < — Dat sin—+ cos— + , V \ 2x \-2x Phuang trinh da cho tra t = tan—+ cot—t <» 2(1 + sin/) = — r sin t (sinr-l)(sin r + 2sin / + 2) = <»sin/ = l / = —+ A:2;T,yteZ Do < / < ;r, nen ta c6 / = — Suy nghiem cua phuang trinh da cho la x = Vi dugiai9 Giai phuang *) Dieu kien:trinh |x| > 32(x - 4x^ - ] = Xet hai truang hop sau ,0 < / < — Khi phuang trinh da cho tra X > Dat X = 32 - cos/ = ^2^8 < = > ( l - c o s / ) = sin^/ sin^/ sin/ sin/sin/ <=> 8(1 - c o s / ) ( l - c o s ^ / ) = <=> (2cos/-l)(4cos^ / - c o s / - s ) = jt <=>cos/ = - , vi c o s ^ / - c o s / - < vai < / < — 2 Suy sin/= ^ Tu ta dugc X = x < - D a t x = ^ , - - < / < ' sin/ 140 Khi phucmg trinh da cho tra 32 r cos/ -+2 = — r - <:ii>8(l + cosr) = — — sinr suit sin^ t sin^ t <^ 8(1 + cos/)(l - cos^ t^ = 3o (2cos/ + l)(4cos^ t + 2cos/ - s) - <^cos/^ ^ ^ " ^ , vi O<cos/<1 v a i - — < / < Suy sin/ = - Tu dugc nghiem cua phuong trinh da cho x = -3 Vay, phuong trinh da cho c6 nghiem x = + V2T • 4V3 -3 + V2T Bai tap phan 2.1 Bai Giai phuong trinh yjl + yjl-x'^ Bai Giai phuong trinh Bai Giai phuong trinh x + Bai Giai phuong trinh Bai Giai phuong trinh X = x|l + 2VI-A;^ j J 3x x2 = 6V2 x +x + x\l\-x^ = - 2x^ Bai Giai phuong trinh x^ + ^ ( l - x ^ ) ^ = 2x^yll-x^ Bai Giai phuong trinh Sx'* -8x^ +1 = V l - x ^ 141 Huang dan bai tap phan 2.1 1, Dat X = smt,- — <t < — Phwcmg trinh da cho.trd 2? Vl + cosr = sinr (1 + cos <^ V2 cos - = sin r + sin 2t ' n;i t: ^ 3t t <»,V2cos— = 2Sm—cos- o 2 2 cos— = 3/ sin— ^ V2 KSt hop voi di€u kien cua t suy / = — hoac t = — Suy nghi?m cua f^tiong trinh jc,= — hoac x = - n• u > Di€u kien: x > DSt x,- ^ , / e sin/ I 0;—2 Xet hai truong hop: Tncomghap J: te Khi phuong trinh da cho tra ^1 ^ — sinr cos/ , smt = - sin t -4 sin/ V.4 COS/ + I ^ ,^ ^ <^ = cos/(cos/ +1) sin2/ = - 4sin/ Doi chieu dieu kien ta c6 /, = — /o = — ' 12^ 12 Truong hap 2: t' ^1, — smt v4 <=> Khi phuong trinh da cho trd cos/ , - s i n / = - Sin / -4 sin/ • + cos/(cos/ - ) = <=> sin2/ = — 4sin/ 142 Doi chieu dieu kien ta c6 t-, —,tA ~ 12 Tir suy nghi?m cua phuang trinh da1 cho X = ,x = -,x\=sin , x = — sinsin 12 sm-12 12 12 Dieu kien: |x|>3 ^ 12^ Tir phuang trinh da cho ta c6 x > Suy x > Dat x = Phuang trinh da cho tra - ^ + ^ ^ = 2V2 CGSt smt cost ,t e <=>sinr,+ cos/ = 4l€\nlt "sin2/ = l <=>l + sin2r = 2sin^ 2/<z> sin2/ = -^i' Doi chieu dieu kien cua t ta c6 t - — Suy nghiem ciia phuang trinh da cho x = 3V2 Di6uki§n: x;t±l Dat x = tanr,/e 2'2 sin/ Phuang trinh da cho tra cos/ cos/cos 2/ <::>COS2/ = COS/ — K2 / 2/ = J cos / = sin/ t^kln = - —+ / + A;2;r D6i chiiu digu kien cua t ta c6 / = — Suy x = ^ V3 Dieu kien: 2/ Dat x = sin/, n k27t t =—+ / = — + k2n: < x < — 143 Khi phuomg trinh da cho tra 2f = cos^ It - + sin ^ cos ? = - sin^ / o 1- + 1- sin 2 sin2^ = - l <:> sin^ 2? + sin 2^ - = ma • ^ 1• sin 2/ = — Doi chieu dieu kien cua tia.c6 u —,U = — ' ^ 12 Tu suy nghiem cua phuong trinh da cho x = ——,x = sin—, Digu kien: |x| < Dat X = c o s / , / E [0;;r] Khi phuong trinh da cho trd cos-' t + sin^ t = cos^ / sin / (1) Ro rang cost - khong thoa man (1) Do (1) tuong ducmg voi tan^ / +1 = tanf <=> ( t a n / - l)(tan^ r + tan/-1) = 'tan/ = l tan/ = Doi chieu dieu kien cua / ta c6 cos / = -7=,cos / -, V2 \5±^/5 Tir suy nghiem cua phuong trinh da cho JC = V2'^ Di6u kien: |x| < Dat JC = cos/, / e [0;;rl ta dugc 7= cos 4/ = sin / <=> cos 4/ = cos 4t = — ~t + k27r At^-— + t + kln i5±y[5' / 71 kin t =— + 10 7t 144 /=— + kin Doi chieu dieu kien cua t ta duac U- — J - , = —,t-.• ' — 2^ 10^ Tir suy nghiem cua phuong trinh da cho 10 ' 2.2 Phuffng phdp ddnh gid bdng ham so V i du Giai phuomg trinh 2UlSx'^ + l0x + - V - x ) - yfx + l LorigidL*)Qihukien:-\<x<3 Phuomg trinh da cho tirorng duorng v a i V x T T + 2>/3-x =27(5JC + ) ^ + (1) Xethams6 / ( x ) = Vx + l + V - X , - l < x < Tac6/'(x)= /•(x) = « — = = , - l < x < -l<x<3 V ^ =2 V m ^ ^ = 2V^ Taco / ( - l ) = , / ( ) - , / Suy / ( x ) < 2Vs < 2^(5^ + ) + , V x e [ - ; ] (2) T u ( l ) v a (2) t a c o V ^ + 2V3^ = 2N/5 2V(5x + l)2+5 = 2V5 o x = — Vay, phuomg trinh da cho c6 nhdt nghi§m = • Vi du Giai phuomg trinh >/2 + x + V - x = x^ - 3x I<W^mA * ) D i k kien: - < x < x> f Xet ham so / ( x ) = x^ - x tren - < x < Taco /•(x) = x - = « x =l x =- l 145 va / ( - ) = / ( I ) = - , / ( - l ) = / ( ) = Suy <2,VXG[-2;2] (1) Matkhac (^y/l + x+y/2~xf =4 + 2yl4-x^ > Suyra V2 + x + V - x >2,Vxe[-2;2] (2) V2 + x + >y2-x =2 Tir(l) va (2) ta c6 «x = +2 x^-3x = Vay, phuong trinh da cho c6 nghiem x = + x^+2x^-2 Vi du Giai phuong trinh x + Vl-x^ = x+2 L&igidL *) Dieu Icien: - < x < Xethamsd / ( x ) = "^^^^ ~^ = x^ ^tren-l<x<l x+2 x+2 'x = - l -3 + V5 Taco / ' ( x ) = 2x + ; (x + 2r Bang bien thien cua / ( x ) : ;/'W = X fXx) fix) 0o -1 X= - = ^ 0 + -1 1 Xet hai truong hop sau a) - l < x < T a c / ( x ) < - l < x + V l ^ ^ , V x e [ - l ; Dau dang thuc xay x = - b) < x < l Chuyrang (x + V l - x ^ =l + 2x>/l-x^ > , < x < l nen x + Vl-x^ >1 146 Vay, fix)<^<\<x + yl\-x^ , Vx e [ ; l ] Suy phuong trinh khong c6 nghiem truong hop Vay, phuong trinh da cho c6 nghiem nhat x = - V i du Giai phuong trinh x +5 + Vx + =1 x'+5 L&igidL * ) Dieu kien: x > - Xethamsd / ( x ) = ^ l i = , x > - x'+5 T a c o fXx) = -p==jL; / ' ( x ) = « x = Ta CO / ( - ) = 1, / ( I ) - V6, l i m / ( x ) = Suy / ( x ) > 1, V x > - D o x +5 + Vx + > l , V x > - Dau dang thuc xay k h i x = - Suy phuong trinh da cho c6 nghiem x = - V i du Giai phuong trinh 3^12-3x2 = X L&igidL + 18 2+2x + l * ) Dieu kien: - < x < , x ; t - - i - Phuong trinh da cho dugc viet lai x + yjl2-3x^ = 2x + l + 2x + l (1) X e t h a m s6 / ( x ) = x + > / l - x ^ , - < x < Tac6/'(x)-l- 3x .,-2<x<2 3x^ ^-2 /•(x) = « < X< 4^-3x^ = 3x <^x = l 147 Taco / ( - ) = - , / ( ) = , / ( l ) = Suy - < f{x) < 4, Vx e [-2; Xet hai truomg hop sau a) — < x < K h i 2x + l + — >6 2x + l Vay,taCO sfX + ^\2-?>x^ < < 2x + l + 2x + l D i u ding thuc xay x = b) - < x < - - K h i d 2x + l + — ^ < - 2x + l Vay, ta c6 S^x + V l - x ^ j > - > -12 > 2 x + l + - 2x + l Suy phirorng trinh khong c6 nghiem truomg hop Tom lai, phuomg trinh da cho c6 nghiem nhdt x = Vi du Giai phuomg trinh + , ^ = >/3 + x - x ^ V2-X Vx Led giai *) Dieu kien: + 2x-x^ >0 <=>0<x<2 t lib ;v 0<x<2 Xet ham s6 / ( x ) = - ^ + J-—,0<x<2 Vx V ^ Taco /'(x) = ^ + —=L_,0<x<2 2Vx^ 2yli2-xf '0<x<2 o x =l Taco / ( I ) = 2, lim / ( x ) = lim / ( x ) = +oo Suyra /(x)>2,Vxe(0;2) (1) « Matkhac, V3 + 2x-x^ = V - ( x - l ) ^ <2 h (2) 148 Tu (1) va (2) suy phuong trinh da cho tuong duong vol yl3 + 2x-x^ = Vay, phuong trinh da cho c6 nghiem nhat x = \ Vi du Giai phuong trinh 3(V2x + V - x ) = x^ -3x^ +13 LdigiaL *) Dieu kien: < x < Xet ham s6 u{x) = yflx + V - x , < x < Taco w'(^) = - l = V2x 2V3-X , < x < ; u'(x) = 0<=>x = Ta CO u(0) = ^/3,M(2) = 3, w(3) = V6 Suy M(X) < w(2)-',=t 3,Vx :» , I ' e! [0;3] Dodo 3(V2^ + > / ^ ) < , VXG[0;3] (1) Mat khac, xet ham s6 v(x) = x^x -3x^ +13, < x < = ^^-^i-Ml^^^ Taco v'(x) = 3x^ - x = 0<=> x=2 Tinh cac gia tri v(0),v(2),v(3) ta dugc v(x) > v(2) = 9,Vx G [0;3] (2) Tu (1) va (2) ta dugc nghiem cua phuong trinh da cho la x = Bai tap phSn 2.2 Bai Giai phuong trinh Vx + + 2yj3-x = -x^ + x - Bai Giai phuong trinh 2A/I2 + X^ + - X = 3^5+ x - x^ Bai Giai phuong trinh 3^A-x^ + = x + 36 5x-2 Bai Giai phuong trinh , , = Vx + + y]3 + 2x-x^ Vx + l + V - x Bai Giai phuong trinh ^"^."'"^ + >/2x-3 = Huong dan bai tap phan 2.2 Dieu kien: - < x < Xet ham s6 f{x) = yjx + l + 2>/3-x tren [-1;3] ta dugc 2</(X)<2V5,VXG[-2;3] Tudotaco + + 2V3-X > > - ( x - ) ^ = - x ^ + x - Dau dang thuc xay x = i Suy nghiem cua phuong trinh da cho la x = Dieu kien: - < x < Xet ham s6 / ( x ) = 2Vl2 + x^ + - x , - l < x < Taco / ( x ) = 2Vl2 + x^ + x - , < x < 2Vl2 + x^ - x + , - l < x < 2x Suyra/'(x) = + 1, < x < V12+? 2x ; /'(x) = « x = -l,-l<x<3 V l + x2 Tinh cac gia tri / ( - l ) , / ( ) , / ( ) , / ( ) tathudugc <9 Mat khac, 3V5 + x - x ^ = 3^9-{x-if (1) / ( X ) > / ( ) = 9,VXG[-1;5 (2) Tir (1) va (2) suy phuong trinh da cho c6 nghiem x = Dieu k i e n : - < X < 2,X ?t - Viet lai phuoTig trinh da cho x + V - x = x - + ^5 x - (1) Xet hai truong hgp: ' j a) - < x < Xetham s6 / ( x ) = x + V - x ^ , - < x < thudugc ^ 150 (1 ' =10,Vxe - ; /oN f{x)<f - 36 Trong x - + 5x-2 > - = 10, dau dang thuc xay x = - g Suy ra, truong hop phuong trinh c6 nghiem ^ = ^• b) -2 < X < — Xet tuong tu truong hop tren, truong hop ta thu dugc U^-x'^ > - > - > x - 5x-2 Nen truong hop Ichong c6 nghiem Dieu kien: - < x < Viet lai phuong trinh da cho - V + x - x ^ = Vx + Vx +1 + V3 - ~H at VxTT + V - X = t,2 < t <2V2 Suy S + 2x-x^ Dat Khi ^ , ^ - V + 2x-x^ Vx + l + V - x Xet ham s6 f{t) = Of X t t = — + 2 + 2, < r < 2V2 ta thu dugc f{t) < f(2) = i_ M i l ' d Vay, ^ -V3 + x - x ^ < l < V ^ , V x g [ - l ; ] A Vx+1+v3-x Dau dang thuc xay x = - D ^}'\ S.DiSukien: — < x < • ^ XethamsS / ( x ) = -iTi'ih +V2x-3,-<x<3 l+v3-x Ta CO / ( x ) dong bien tren doan 1^^ ;v a '^J;ni , ^ Xet cac truong hop sau +) - < X < Khi fix) < / ( ) = 2, dSu dang thuc xay x = +) < X < Khi / ( x ) > / ( ) = 2, nen phuong trinh khong c6 nghiem truong hop Vay, phuong trinh da cho c6 nghiem nhSt x = 2.3 Phuangphdp ham s6 dan dieu ., - - ; (4 Phuong phap ham so don dieu dua tren cac ket qua co ban sau day: Neu ham so y = f(x) don dieu tren K thi phuong trinh / ( x ) = c6 khong qua nghiem T6ng quat: NSu f'(x) lien tuc va c6 n nghiem tren K thi phuong trinh / ( x ) = CO khong qua « +1 nghiem tren Ậ N6u ham so >- = / ( x ) d6ng bi§n, ham y = g(x) nghich bien tren K thi phuong trinh / ( x ) = g(x) c6 khong qua nghiem tren K Ngu ham s6 f (x) don dieu tren K thi phuong trinh ^ •" / ( " ) = / ( v ) <=> w = V voi M,V G/iT Phuong phap ham so don dieu thuong dugc su dung de giai cac phuong trinh CO mot hai dang sau day: Dang Phuong trinh / ( x ) = 0, chung ta c6 ihk lap dugc bang biSn thien ciia ham y-f (x) Dang c6 cac truong hgp dac biet: - Ham y = f(x) don dieu tren khoang D k - Ham y = / ( x ) don dieu tren moi khoang c D, u A", =D va cac khoang 1=1 AT,, roi r; - Phuong trinh / ( x ) = <=> g(x) = h{x), >' = g{x) va j = h{x) la cac ham don dieu ngugc chiSu tren D Mot yeu cau can thiet giai phuong trinh Dang la chung ta chi tinh don dieu cua / ( x ) tren cac khoang KciD, rhi nhkm tim dugc nghiem cua phuong trinh tren K (neu c6), sau su dung cac tinh chit da trinh bay we ham m dcm dieu de ket luan la tat ca cac nghiem cua phuong trinh da cho Truong hop phuong trinh khong c6 nghiem tren K thi dua vao dac diSm / ( x ) trenATdg kSt luan gia tri ciia , '.^ Dang Phuong trinh / ( x ) = F ( w ( x ) ) = F ( v ( x ) ) , y = F{t) la ham don dieu tren K va w(x), v(x) thuoc K voi moi x G D Khi do, ta CO F ( M ( X ) ) = F ( V ( X ) ) <^ u{x) = v(x) 6-x V i du Giai phuong trinh 1+Vl-x - + VX + LofigiaL *) Didu kien: < x < Phuong trinh da cho tuong duong voi (1) Xet ham so / ( x ) = 4~x - + Vx + , < x < l 1+>/r~x .[^•-t J<% d a £1 CM} Nhan th4y / ( x ) d6ng bien tren [ ; l ] va / ( I ) = Do phuong trinh (1) tro / ( x ) = / ( I ) x = Vay, phuong trinh da cho c6 nghiem x = Vi du Giai phuong trinh Vx^ - x + + V x - = LdigiaL *) DiSu kien: x > Xet ham s6 fix) = Vx^-x + + V2x-1, x > - „ r ^ 2X-1 ^ Taco/'(y)= , + , >0,x>- 2Vx^-x + V x - Suy / ( x ) dong bien tren khoang ' Chu y rang / ( ) = Do phuong trinh da cho tuong duong voi / ( x ) = / ( ) « x = Vay, phuong trinh da cho c6 nghiem x = 153 Vi du Giai phuang trinh + = =x+1 + Vr^~ \ ^2(\-x) L&igidi *) Dieu kien: - < x < Xet ham so /(t) = — ^ ^ = , <t<4 l+V4-r T a c o / ' ( = -^ ^'^O'O^^^'^ V4^.V^(l + V ^ ) Suyra f(t) ddngbilntren [0;4] Phucng trinh da cho dirge viet lai dang / ( x + ) - / ( x + 2) = x - l Chii y rang, x < nen x + > 2x + Suy fix + 3) > / ( x + 2), hay f(x + 3) - / ( x + 2) = x - > Din din Do ta CO X = , , Vi du Giai phuang trinh 4x^ + 5x-(x + 3)V2x+T = LMgiai *) Dieu kien: x > - Xet ham so f{t) = + 5/,r G M Ta c6 f{t) la ham d6ng hxkn tren M Phuang trinh da cho dugc vigt lai dual dang / ( x ) = / ( V x + l)<:^2x = V2x + l c>x = - ' - i ^ Vay, phuang trinh da cho c6 nghiem x = ^ ^ ^ N\u Giai phuang trinh 4x^ + x - (x + l)V2x + l = LM giai Dieu kien: x > - ^ Phuang trihh tuang duang vai (2x)^ + 2x = (72x4-1^ + V2x + (1) 154 Xet ham s6 f(t) = P+t tren R Ta c6 f'(t) = 3t^ +1; f \ t ) > vai moi t eM Suy ham /(t) d6ng biln tren R Do (1)«/(2x) = / ( V x + l)«2x = V2x + l ' " r2x>0 i + v^ [4x^=2x + l x=^^^ Vi du Giai phuong trinh (x^ -l)V2x + l 4x' LM giaL *) Dieu kien: X > - ^ Vay nghiem cua phuong trinh la x +\ x^+\ - r,: Chu y ring, tir phuong trinh da cho ta c6 x > i Phuong trinh da cho dugc viet lai dudi dang ' A' x^-l^(2x + l)^-l X V2x+1 Xet ham s6 / ( O = Ta CO / '(0 = .- , • > + ^ > 0, nen f{t) d6ng biSn tren khoang (O; + oo) Do phuong trinh (1) tuong duong voi / W - / ( V x + l)«x = V2x + l ^ x = l + V2 ^^jmx ; • ) \ Vay, phuong trinh da cho c6 nghiem x = + ^/2 Vi du Giai phuong trinh x'' + 3x^ + 2x = 2(x + l)V2x + L&igiaL *) Dieu kien: x > - ^ Phuong trinh da cho tuong duong voi (x + l ) ^ - ( x + l) = (V2x + 3)^-V2x + (1) 155 - Xethamso^ f{t) = t-t,t> Taco f\t) = 3r-\>0^ ^ ^ ; / ( o ) = / ( i ) = o Bang biln thien cua /(t): /(O _1 + 0 - V30 +C0 + < Xet cac truang hop sau: <x+l<0 a) — < x < - l K h i d / ( x + l ) > > / ( V x + 3) 0<V2x + <1 Dau dang thiic xay x = - J" b) - l < x < K h i d < x + l<l<>/2x + 3=:^/(x + l ) < < / ( V x + 3) Suy phuong trinh khong c6 nghiem truong hop VH}'v c) x > Khido x + l>l,V2x + > Do /{t) dong bien tren khoang ; + co , nen phuong trinh (1) tuong duong voi / ( x + l) = / ( V x + 3)<z>x + l = V2x + c^x = V2 Tom lai, phuong trinh da cho c6 nghiem x -l,x = -Jl Vi du Giai bit phuong trinh 3^3-2x + 2x < V ^ <X LaigiaL Digu kien: — < x < — ' \1 Xet ham s6 /(x) = 3V3 - 2x + V x - - x - tren 2' -Vl-xV 156 Taco f\x) = -3 J - < voi moi x e ( \^ 2' (V2^} Do ham f{x) nghich biSn tren 1 2'2 Ta thay / ( I ) = nen bit phuong trinh tuong duong voi f{x)<f{\)<:>x>\ Ket hop dieu kien ta suy nghiem cua bit phuong trinh la < x < — Vi du Giai bit phuong trinh V3x + > (1 - x)^ + L&igidL Dieu kien: x>-^ Bat phuong trinh da cho tuong duong voi >/3x + l - ( l - x ) ^ - > (1) Xet ham s6 / ( x ) = >/3x + l - (1 - x)^ - tren doan • • Taco / ' ( x ) = -;^-^==== 2V3X + + ( l - x ) ^ > voi moi X E Do ham so / ( x ) d6ng biSn tren Mat khac, ta thiy / ( I ) = nen suy ( ) « / ( x ) > <=> x > Vay nghiem cua bit phuong trinh la x > Vi du 10 Giai phuong trinh Sx^+13 = 4x - + >/3x^+6 L&igidL Xet cac trucmg hop sau Neu X < - thi 4x - < 0, do FT > VP nen phuong trinh v6 nghiem Neu X > —, thi phuong trinh da cho tuong duong vol ' • ' V ? + - V ? + - x + = Xethams6 / ( x ) = V3x^+13 ->/3x^+6-4x + voi x > ^ 157 Taco f'[x) = 3x - 4; f'(x) < vai moi x > - Suy ham / ( x ) dong bien tren khoang Mat Idiac / ( l ) = do x = la nghiem nhat cua phuong trinh / ( x ) = Vi du 11 Giai phuong trinh 2V4x^ - x + + 2x - 3^12x^ -x^ + V9x^ - x + (HSGNgheAn 2007) LM giai Phuong trinh xac dinh vai moi x e M Ta thay x = la mot nghiem cua phuong trinh Xet X > 0, chia2V?^-r hai ve phuong trinh + + =trinh ^ 2cho ^ - x+ Vva4 fdat ^ - 4/ ^= +—9 ta dugc phuong(1) phuong trinh (1) trothanh Dat u = y}lt-\, V W ^ + + = 3M + VM^+8«3M + N / M * + - V M ^ + - = (2) Xet ham s6 f{u) = 3u + - V w ^ + - tren R Ta thiy nSu w < thi /(«) < nen ta chi cin xet w > Khi /•(M) = + 3M= V ? + V w ^ + 15 >0 voi moi w>0 Do f{u) la ham dong bien tren (O; + oo) Ma / ( I ) = nen w = la nghiem nhdt cua (2) Tu ta tim dugc x = Xet X < 0, lam tuong tu nhu tren ta c6 phuong trinh V W ^ + - M - V W ^ + + = voi Xet ham so g{u) = Taco g\u) = 3u- VM^+8 + 15+2 voi u<-\ u<-\ - <0 voi moi VM^ + 15 u<-\ 158 Neng(w)>g(-1) = ' Vay X = 0, X = la nghiem cua phucmg trinh da cho Vi du 12 Giai phuang trinh fc + 1-3 x + 2' (HSGNgheAnlOB) fx>-l L&igidL Dieu kien: [ x ^ l Phuang trinh da cho tuomg duong voi (x + ) ( V ^ - ) = fc + l - (x +1)Vx+T + = 2x +1 + ^2x +1 (1) Xet ham s6 f[t) = t^+t vai t eR Ta c6 / ' ( / ) = 3/^+1>0 vai moi / e R Suy ham so / ( / ) dong bi8n tren M Khi (1) o / ( V ^ ) = / ( ^ x + l) o x> — (x + l / = ( x + l)^ x.-l = + '' ' ' ' ' x.-i x=0 ^ « i + Vs x3-x'-x =0 x= Doi chieu dieu kien ta dugc nghiem cua phuang trinh da cho la X= va X= x=0 + N/5 X = Vi du 13 Giai phuang trinh -2x^ + lOx^ - 17x + = 2x^^5x-x^ L&igidL Ta thay x = khong la nghiem cua phuang trinh Vai X thi phuang trinh da cho tuang duang vai X x^ x^ -1 159 Dat t = -, t^O YSLphuofngtrinhtrathanh X <:> (2t - \f + 2(2/ - ) = 5/' - + 2^5/^ - (*) Xet ham so f{u) = u^+2u tren R Ta c6 / ' ( « ) = 3u^ + 2; f'(u) > vai moi X= Suy Suyraham / ( w ) dong biSn tren R Do MGE (*) o f{2t - ) = f{^^5t^ - ) c^i2t-lf 2/ - = ^ / ' - ^5t^-Ic^Sp-nt^+6t =0 't = (Ictm) / = 17±V97 16 16 Vay nghiem cua phuomg trinh la x = 16 17±V97' Vi du 14 Giai phuang trinh 3xl2 + V ? + ) + (4x + 2)(VT+xT? +1) = L&igidL Ta c6 phuang trinh da cho tucmg ducmg vai r i^.)] (4x + ) | V l + + ? + l j = -2 -3x + y]9x^+3 <=> (2x + l)(v^4 + 4x + 4x^ + = - x + V ? + ) (2x + l)(V(2x + ) H + 2) = (-3x)(V(-3x)^ + + 2J (1) Xet ham so f(t) = t{ylt^+3 + 2J vai / e R I Taco / ' ( = + V / ^ + + - _ ; / ' ( > vai moi x e R V/'+3 Do ham f(t) ddngbilntren R t^^ 160 Suyra (1)<» / ( x + l ) = / ( - x ) < : ^ x + l = - x < : ^ x = - ^ Vay nghiem cua phuomg trinh la x = V i du 15 Giai bat phuomg trinh >/x^-2x + - V x ^ - x + l l > V ^ - V ^ x^-2x + 3>0 L&igidi *) Dieu kien: - x ^ - x + l l > C:>1<X<3 (1) 3-x>0;x-l>0 Ta CO bat phuomg trinh da cho tuong duong voi Vx^ - 2x + + V x ^ > Vx^ - 6x +11 + V J ^ X ^V(^-l)^+2 + V ^ > V ( - x ) ^ + + V ^X Xet ham s6 f{t) = \lt^ +2 +yft wai (2) t>0 Ta CO fXt) = -fJ— + ^;f'{t)>0 t' + 24't' vai moi t>0 Do ham s6 f{t) dong bien tren [O; + oo) Matkhac x - va - x d^uthuoc [O; + o o ) Suyra (2)«/(x-l)> /(3-x)c:>x-l>3-xc^x>2 Doi chilu vai diSu kien (1), ta c6 nghiem cua bdt phuomg trinh la < x < Bai tap phan 2.3 Bai Giai phuomg trinh ( x - ) —, =x^— Bai Giai phuomg trinh V^ + l + >/x^ + x + l -•^2x + 4^x^ - x + l = x - Bai Giai phuang trinh 8x^ -17x^ +1 Ox - = f e ^ - l Bai Giai phuang trinh 4x^ +18x^ + 27x +14 = 1[AX + Bai Giai phuomg trinh yJx + + V - x = x^ - x + 161 Bai Giai phiromg trinh - 3x +1 = V s - S x ^ Bai Giai hk phiromg trinh Vx + + x^ - x - < V x - Bai Giai phiromg trinh 4Vx + + V 2 - x = x^ + Bai Giai phuorng trinh 8x^ -36x^ + 53x- 25 = >/3x-5 Huang dan bai tap phan 2.3 \ Phuomg trinh da cho tuomg diromg voi (5x - 6)' - , ^ = x^ ; ^ V(5x-6)-i Xetham s6 f(t) = t^ T a c o / • ( r ) = 2r + — -jL= J — voi ^ > ; / ' ( > Q vai moi r > l 2V(^-1)' Suy / ( O dong bien tren ( l ; + oo) Do phuomg trinh da cho tuomg duomg vai / ( x - 6) = / ( x ) <^ 5x - = X c:> X = ^ (thoa man) Vay phuong trinh c6 nghiem nhat ^ = ~ • Dieu kien: : i-'X x + l + >/x^+x + l > ^2x + >/4x^-2x + l>0 Phuomg trinh da cho tuomg duomg vai x + l + ^x + l + V(x + l)^-(x + l) + l =2x + ^2x + ^(2x)^-2x + l Xet ham s6 f(t) = t + \jt + yjt^ -t + \n tap xac dinh Taco •f + l + ? - l =1+ = !+• 162 Dc y rang ::' „ x 2yjt^ -t + \+ 2t-\^yl4t^ -4t + , , ,^ 4+2t-\ =V(2/-l)^+3+2/-l>|2/-l| + 2f-l>0 Suy f\x) > vai moi x thuoc tap xac dinh Do ham s6 d6ng biSn tren tap xac dinh Khido (!)<» + /(2x)<=>x + l = 2x<::>x = l , thoa man diSu kien Vay nghiem ciia phucmg trinh la x = Phuong trinh da cho tuong duong vai (2x -1)^ + 2(2x - ) = 5x^ - + 2^5x^ - (*) Xet ham so / ( / ) = t^ + 2t tren R T a c o / ' ( = / ^ + ; / ' ( > vai moi r e R ' ' Suy ham / ( / ) d6ng bi^n tren R Do (*)<=> / ( x - l ) = / ( f e ^ - l j < ^ x - l = fe^-l "x = (2x -1)^ = 5x^ - « 8x^ -17x^ + 6x = 17±V97 x= Vay nghiem cua phuong trinh la x = 0, x = 16 17 + V97 16 Phuong trinh tuong duomg vai (2x + 3)^ + 2(2x + 3) = 4x + + 2^4x + ^ (2) Xet ham s6 f(t) = t^ + 2t tren R Ta c6 f\t) = 3/^ + > vai moi t eR Do ham / ( / ) dong bien tren R Suy ( ) o / ( x + 3) = / ( ^ x + ) < » x + = ^ x + <:>(2x + 3)^=4x + o x ^ + x ^ + x - 2 = "x = - l < » ( x + l)(4x^+14x + l l ) = o -7±V5 x= 163 Vay nghiem cua phuong trinh la x = -1, jc = Digulcien: - + V5 -2<x<3 Phuong trinh da cho tuong duong voi y/x + 2+y/3-X-X^+X-l Xet ham s6 f{x) = yJx + + ^3-x = -x^ +x-l tren doan [-2; Ta CO f\x) = V ^ voi moi XG(-2; ^ -2x + l;/"(x) = ^ 2V3^ ' 4^(x + 2f 3) Do phuong trinh f\x) 4^1 -2<0 (3-xf = c6 t6i da nghiem, nen phuong trinh f(x) = c6 khong qua nghiem Mat khac, ta thay x = 2, x = -1 la nghiem cua phuong trinh / ( x ) = Vay nghiem cua phuong trinh la x = 2, x = -1 Digu Icien: - 3x^ > Khi phuong trinh da cho tuong duong voi x ^ - x + l - ( - x ) + ( - x ) - V - x ^ =0 (x + l ) ( x ^- x- x- -1l ) + X + -4(x'-x-l) ^ ^ (2-x)-V8-3x' 1+ ( - x ) + V8-3x^ Ta chung minh phuong trinh x +1 + =0 = (1) = (2) v6 nghiem ( - x ) + V8-3x^ Xet ham s6 / ( x ) = - x + V8-3x^ vol x e Taco / ' ( x ) = - l - '3'V3 3x ^ ^ ; / ' ( x ) = 0<::>x = - - ; / V8-3x' V3 + 4>/6 V3 164 / Do = 2- X +1 = + ^, ^ > + fix) >1- SuyraO</(x)<^±^ > Nen phuomg trinh (2) v6 nghiem +V3 + 4^/6 Tu suy phuang trinh (1) «> - x - = <=> x = Vay nghiem cua phuang trinh la x = 1±V5 1±V5 DieuIcien: x > — Khi bat phuang trinh da cho tuang duang vai ( V ^ - V x - ) + (x'-x-2)<0 2(2-x) V ^ + (x-2)(x + l ) < + V3x-2 <»(x-2) -2 + X + ^yxT2+^/3x-2 <0 (1) -2 , — = + x + l vai x > — Xet ham so / ( x ) = ; Vx + + V3x - 3 + Taco/'(x) = l + ^ - ^ " ^ ^ — — ^ > vai moi x > - (Vx + + V x - ) ^ Do ham / ( x ) dong h\kn tren Suyra / ( x ) > / ^ — — > vai moi X > — V2 - Suy bit phuang trinh (1) <=> x - < o x < 2 Ket hap di6u kien, ta c6 nghiem cua hk phuang trinh la — < x < 165 Dieukien: -2<x< 22 Khi phuong trinh da cho tuong duong voi 4(V]rh2-2) + ( V 2 - x - ) = x ^ - V22-3X + y/x + 2+2 3(2-x) 4(x-2) < z > ( x - ) x + 2- , yJx + 2+2 =0 V22-3x + x=2 x + 2- > / ^ +2 Xethams6 / ( x ) = x + CO /'(x) =0 voi - < X < 22 V22-3X+4 =l+ (1) V22-3X+4 + - Vx + 2+2 Ta > voi moi V X + 2(N/X + + ) " ^ V 2 - X ( N / 2 - X + ) xe -2; 22 Do ham / ( x ) dong bien tren - ; 22 3^ Mat khac, ta thay / ( - I ) = nen x = -\a nghiem day nhat cua (1) Vay nghiem cua phuong trinh la x = 2, x = - Phuong trinh da cho tuong duong voi 8x^-36x^+54x-27 + 2x-3 = 3x-5 + <^(2x-3)^+2x-3 = 3x-5 + fc-5 (2) fc-5 Xet ham s6 f(t) = P +t tren R Ta c6 / '(t) = 3^^ +1 > voi moi / G R Do ham / ( t ) dong bien tren R Suy (2) / ( x - 3) = / ( ^ x - ) o 2x - = ^3x - «8x^-36x^+51x-22 = 0o(x-2)(8x^-20x +l l ) - 166 X= + V3 Vay nghiem cua phuong trinh la x = 2, x = + V3 3) Vi du tong hgfp Chxrong Vi du Giai phuong trinh Vx + Vx + + V ? + x + 2x = 35 L&i giai *) Dieu kien: x > Dat V^ = a;Vx + 7=6,(a,6>0) Tir phuong trinh da cho va each dat in phu ta dugc a + b + 2ab + 2a^ ^25 Ua + b){\ 2a) = ?,5 ^\{b + a){b-a)^l : Tudotaco l + 2a = 5(6-a),hay 7a + l = 56 Dieu tuong duong voi 7V^ + l-5>/x + 49x +1 + lAsfi = 25(x + 7) 1^ = 87 - 12x < x < 87 12 144x^-2137x +7569 = ,v6 nghiem Vi du Giai phuong trinh ( l - V l - x ) ^ - x = x LofigiaL Dieu kien: x < Phuong trinh tuong duong voi X ^ - X =X(1 Giai (*) Dat u = u+1 =v v^-u^=l + Vr^) <:> x = ^ - x =l + Vroc v = ^ - x , u > Khi ta c6 he u = v-1 [v^-(v-l)^=l (*) u = v-1 [v^-v^+2v-2 = 167 fu = v - l fv = l (v-l)(v^+2) = [u = Suy x = Vay nghiem cua phucmg trinh la x = 0, x = Vi du Giai phuong trinh (Vx + ^ x>l / I - ) yjx^-x+ \ J Y ^ — = L&igidL *)Dieulcien: Phuong trinh da cho tuong duong voi <^ylx -x+ I =- = - V x < z > + Vx-1 x-1 =t>\ Khi phuong trinh da cho tro Dat x-1 + ^2 ^ ^ < = > ( r - ) ( / ^ + ^ - l ) = 0«/ = 2, vi 2t^ +4t-\>0yt>\ Suy nghiem cua phuong trinh ban dku ^ = • Vi du Giai phuong trinh - = j = = V x - + >/3-x + V3 + x - x ^ Vx+1+V3-X L&igidL *) Dihu kien: - < x < Viet lai phuong trinh da cho -P=^ ; -V3 + 2x-x^ = V I ^ + V ^ Vx + + V ^ ^2-4 Dat V x T I + V - X =t,2<t< lyfl Suy V3 + x - x ^ = Khi , ^ , -V3 + 2x-x^ = - - — + Vx + l + V - x ? 2 Xethamsd fit) = - - — + 2,2<t<2^ tathudugc / ( < / ( ) = l + 2x-x^ <l,Vxe[-l;3 (1) Vay, ; ^ I—^-V3 VX+1+V3-X 168 Mat khac ( V ^ + yJJ^f = + l^I^.yfJ^ > Suy +V3^>1,VXG[-1;3] (2) T u ( l ) v a (2) t a d u g c slx + l+yj3-X Dau dang thuc xay k h i x = Suy nghiem cua phuong trinh da cho la x = V i du Giai phuomg trinh Vx - x + + ylx^+lx + l = ^ L&igidL *) D i e u kien: x > Nhan thay x = khong phai la nghiem cua phuomg t r i n h da cho, v i vay x > K h i phuong trinh t u o n g duomg v a i x + 1+ X Dat x + - = r > t a d u o c y[t^ + ^t + l ^ V X + -X + = =A C h u y rang ham s6 / ( t ) = V ^ - + -Jt + l d6ng b i l n tren khoang [2; + oo) v a / ( ) = n e n t a c o 4T^ + 4t + l =A<^ f{t) = f{2)<^t = T u ta thu dugfc nghiem cua phuong trinh da cho la x = V i du Giai phuong trinh (x^ + x)^ + (x - ) ^ = (x^ + l ) V x - x ^ LMgiaL * ) DiSu kien: x - x^ > « x<-l 0<x<l Phuong trinh da cho t u o n g duomg v o i (x^+l) Dat 4^ x^+l -2(x-x^) = (x2+l)Vx-x^ « l - x-x^ (x^.l)2 yjx-x^ x^+l / =- l = t ta duoc r + / - = <:> 169 Suy 2ylx-x^ =x^ + 4x-—^ = \ x^ (1) \ x' Dat x = tana.Khid6(l)trathanh sin4a = l,hay a = ^ + ^,keZ Doi cliieu dieu kien ta chon cac gia tri ciia k\a k = -l,k = 1; Suy nghiem cua phuong trinh da cho x = tan—, x = t a n ^ ^ 8 Vi du Giai phuong trinh ^/x + l + Vx = \/x + Vx + L&igidL *) Di6u kien: x > Phuong trinh da cho tuong duong voi - ^ = VJTT <^^(x+\f + ^ x ( x + i ) + \ / ? = VxTT+V^ Xet cac trucrng hop sau day + ) X = 0, thoa man > Ta CO (1) +) X Suy (1) khong c6 nghiem x > +) < x < l T a c ^x{x + l)>^>4^ Suy (1) khong c6 nghiem < x < Vay, phuong trinh da cho c6 nghiem nhat x = Vi du Giai phuong trinh lOx^ - 9x - 8xV2x^-3x + l +3 = x>l x<2 L&i giai Dieu kien: 2x^ - 3x +1 > 170 TH x<Q khong la nghiem ciia phirong trinh TH x> Phuong trinh da cho tuong duong vol X Dat t= X x' X ! /-IS x' x^ Phuong trinh tro 10 + 3(t^-2)-8t = o t ^ - t + = 0<» t=2 t= * Voi t = , t a c - - + ^ = ^ x ^ + x - l = 0«»x = X Nghiem x = ^^5^mandieukien * Voi t = - , ta CO - - + ^ = -<»14x^-27x + = o X = —2 , thoa man X 3 X=— dieu kien , Vay nghiem cua phuong trinh, la x = -3 XI- + V17 3 , x = —, x =— Vi du Giai bit phuong trinh V4x^+38x-l - y / x - \ + \ 4x^+38x-l>0 <=>x>- ^1 LafigiaL *) DiSu kien: 6x-l>0 Bat phuong trinh da cho tuong duong voi V x H x - l > x + l + 2V6x-l o 4x^ + 38x - > x^ + 2x +1 + 4(6x -1) + 4(x + l)V6x-l o x H 12x + > 4(x + l)V6x - o 3(x +1)^ + (6x -1) > 4(x + l)V6x-l 171 <=> 6x-l ^ V6x-1 V6x-1 „ N/6X-1 ^ , r-4 + > <» >3 hoac <1 x+1 x+1 x+1 (x + i y / < ^ x - l < ( x + l)^ ^ x ^ - x + > < ^ x+1 +) / > ^ x - l > ( x + l)^ ^ x ^ + x + 10<0, v6 nghiem x+1 +) x>2 + V2 x<2-V2 K6t hop diSu kien ta c6 nghiem la x > + yfl, — < x < - V2 Vi du 10 Giai phuong trinh + = yjlx^-3x-4 L&igidL *) D i l u kien: x > Phuong trinh « X + x^ -1 + 27x()?-l) = 2x^ - 3x - <»2Vx(x^-l) =x ^ - x - 2^ x+1 x^-x x+1 x^-x V x+1 lylix^ - x)(x +1) = (x^ - x) - 3(x +1) -3 = ( v i x > nenx + # ) =3 = - (ktm) x+1 <=> x^ - X = 9(x +1) <=> x^ - lOx - = x = + V34 _x = 5-^34 (ktm) Vay nghiem cua phuong trinh la x = + ^34 V i du 11 Giai phuong trinh 2x' + (x-1)Vx+T = 5x + L&i giau *) Dieu kien: x > - Phuong trinh da cho tro (x-l)(VJ+T-2) + x ^ - x - = -Df^xi 172 (x - ) ( - 2) + (x - 3)(2x + 3) = o (x - ) - ^ ^ ^ ^ + (x - 3)(2JC + 3) = Vx+1+2 O(X-3) ,vm+2 + 2X + = +) x = 3, thoaman +) — + 2x + = Vai x > -1 ta c6 yJx + l+2 > - l va 2x + > l Suyra-^li^— + 2x + 3>0 Vx + 1+2 Dau dang thuc xay x = - Tom lai, phuong trinh da cho c6 nghiem la x = -1, x = Vi du 12 Giai phuomg trinh 4(2x' +1) + 3(x^ - x ) V x - l = 2(x^ + 5x) LatigiaL *) Dieu kien: ^ - ~ Phuang trinh da cho tuorng duong voi 3x(x - 2)>/2x-l = 2(x^ - 4x^ + 5x - 2) O 3x(x - ) V x - l = 2(x - 2)(x^ - 2x +1) ., "^ = _ x V x - l = 2(x^ - 2x +1) Phuang trinh (1) tuomg duomg vai 2(2x-1) + x y x - l - 2x' = ^ ^ + ^ ^ ^ ^ - = X l2x-\ Dat t = —-—, t>0 X K h i phuomg trinh (2) tra X / ^ + r - = c ^ ( f - l ) ( / + 2) = c ^ ^ = ^ 2t t>0 Suyrax^-8x + = 0<=>x = + 2>/3, thoamandihukien } Vay nghiem cua phuomg trinh la x = 2, x = + 2^J3, x = - 2>/3 Vi du 13 Giai hk phuomg trinh V4 + x + V l - x < L&igiai Bat phuomg trinh da cho c:> V4 + x < - V l - x <=> V4 + x - < - V l - x V4+I + (2 + Vl6-x)(4 + V l - x ) • (2 + V I - x ) ( + V I - x ) - ( V + X + 2)]< (1) Vai - < x < taco (2 + Vl6-x)(4 + V l - x ) > > V + X + 2, hay (2 + Vl6-x)(4 + V l - x ) - ( V + x + 2)>0 Do bat phuomg trinh (1) tuong duong voi x<0 <::> -4 < X < [-4<x<16 ' a:.;rnEU'itfc, Vay, bat phuomg trinh da cho c6 nghiem la: - < x < Vi du 14 Giai bit phuomg trinh 4Vx+T + 2V2x + < (x-l)(x^ - ) L&igidL *) Dieu kien: x > - Nhan thay x = - la mot nghiem cua bat phuomg trinh ' ' Xet X > - Khi bat phuomg trinh da cho tuong duong voi ' ' M ( V ^ - ) + 2(V2x + - ) < x ^ - x ^ - x - (— x - ) + / 4(x-3) <(x-3)(x^ + 2x + 4) Vx + 1+2 V2X + 3+3 o(x-3) 4 • + V^m + V2X + 3+3 - ( x + i r - <o -(x + l)^-3<0 •+ V ^ + V2X + + VI v a y -1 nen Vi X> +1 > va V2x + > Suy (1) , , <3, V x + + V2X+3+3 Do bat phuong trinh (1) <=> x - > <=> x > Vay nghiem cua bat phuomg trinh la x = -1 va x > Vi du 15 Giai bat phuong trinh 2(1 - x)Vx^ + x - l < x^ - 2x - Ldigiai *) Dieu kien: x + 2x - > <=> 'L If;"-' x > - l + V2 x<-l-V2 Dat v x ^ + x - l =a, I- x-b Khi bat phuong trinh da cho tro 2ab<a^ -4(l-b)^ia-2)ia Ux'+2x-l-2 + 2-2b)<0 Ux^ + x - l + 2x <0 (1) Xet hai truong hop sau +) x > - l + V2.Khid6 V x + x - l + x > Do (1) tuong duong voi V x + x - l - < Giai dugc nghiem - + V2 < x < +) x < - l - V Khi t) mi > / x + x - l + x = V(>: + l ) ^ - + x < | x + l| + 2x = - x - l + 2x = - l + x < ; til Do (1) tuong duong vdi V x + x - l - > Giai dugc nghiem x < -1 - V6 Tom lai, bat phuong trinh da cho c6 nghiem -1 + V < x < l ; x < - l - V Vi du 16 Giai phuong trinh ^6x + l = 8x^ - 4x - , Lo"/g/di Phuong trinh da cho tuong duong voi r,^ 6x + l + ^6x + l=(2x)^ + 2x (1) Xet ham s6 / ( / ) = P +t tren R Ta CO f\t) = 3t^ +1; f\t) > vdi moi t e M Do ham /(t) d6ng biSn tren R 175 Suy (1) < = > / ( ^ x + l ) = f{2x) ^ x + l = 2x o - x - = (2) V o i X e [ - ; l ] , dat x = cos w, w e [O; ; r ] , phuong trinh (2) tra ( c o s ^ M - C O S M ) - = 0<=>COSBM = -<=>3M = ± — + kin, Do wefO; TT] nen M = — = — hoac u = — L ' J 9' • ksZ STT Suy x = cos—,x = cos ^ 9' hoac X - c o s - ^ V i (2) la phuong trinh bac nen c6 khong qua nghiem, do day la tat ca cac nghiem cua phuong trinh (2) Vay nghiem cua phuong trinh la x = cos '^- ^ ' ^~ ' V i du 17 Giai phuong trinh 9x^ - 28x + 21 = V x ^ L&i gidL *) Dieu kien: x > Ta CO phuong trinh da cho tuong duong vai (3x - 5)^ + (3x - 5) = (x - ) + V x ^ Xethamsd (1) f(t)=^t^+t,te Ta CO / ( t ) dong bien tren Neu va nghich bien tren -Qo: thi x - > - — Do x - va V x - cung thuoc Suyra (l)<^ /(3x-5) = / ( V ^ ) c ^ x - = V ^ 3x-5>0 < » x = '^|(3x-5)^=x-l De y rang phuong trinh da cho c6 the viet tuong duong (4-3x)^+4-3x = x - l+ V x ^ (2) Neu l < x < - thi - x > - - 2 176 Dodo ( ) o / ( - x ) = / ( V ^ ) < » - x = V ^ 4-3x>0 25-713 <=>x = (4-3x)^=x-l 18 Vay nghiem cua phuomg trinh la x = 2, x = 25-Vl3 18 Bai tap tong hap Chu'O'ng Bai Giai phuomg trinh ^'^''^ 3yJ\-x^ = l-x2 Bai Giai phuomg trinh Vx^ +1 - V2x^ - x + = x^ - x Bai Giai phuomg trinh 2x +1 + xVx^ +2 + (x + 1)V X + 2x + = Bai Giai phuomg trinh V x + + V2x + = x^ - x - Bai Giai phuong trinh ^(x-lf - -l^JT^-(x-5)V^-3x + 31 = Bai Giai phuomg trinh x^ - 3x = Vx + Bai Giai bat phuomg trinh , ^ = Vx^ + X - Bai Giai bat phuomg trinh > - X ^ ,-^=> V2x-Vx +1 ^ 3-2x •^•"•'^^^ x-1 Bai Giai bat phuomg trinh ( x - ) ( V x T T +1) < x - x Bai 10 Giai phuong trinh sl^fx^ + ^(x-lf ^x'-x Huang dan bai tap tong hop Chuang DiSu kien: -1 < x < Phuomg trinh da cho tuomg duomg vai x ( l - x ^ ) + 2x^ — ^-z l-x^ ^ r = 3V1 - J ^ x^ o X+ , r ^ = 3V1 l-x^ 177 X <^ , X +2 -3 Bat , ^ _ =t tadugc 2p +t-3 = 0^it-l)(2t^ ^ x ' +2t + 3)^0<^t^\ Tir giai dugc ngliiem cua pliuong trinh x = Phuong trinh da cho tuong duong voi x-x -X -X Chu y ring dAu hai v4 cua phuong trinh d6i lap Tu suy nghiem cua phuong trinh x = 0, x = 3.Dat V x ^ + = w > , V x ^ + x + = v > Ta CO - = 2x +1, hay x = Khi phuong trinh da cho tro 1/ ^2 1' (u-v) u + v + -(u + v) +- = 0<=>M = V 2^ ^ Tu giai dugc nghiem cua phuong trinh ^ ~ x>-l <^x>-l Dieu kien: X> — Khi phuong trinh da cho tro V ^ - + V2x + - = x^ - x - , + 2) <^ , x - + ,2(x-3) - = (x-3)(x V ^ + V2X + + Tir day ta c6 truong hgp X = Thoa man 178 + • 2.-== = x + 2.(l) Vx + + V2X + + _Vai ;c > - ta c6 Vx+T + > va V2x + + > Suyra-p=:L + ^ < - + - = l < x + Vx + + V2X + +3 2 _ Do phuomg trinh (1) tuong duong vai Vx+I + V2x + + «>x = - l x+2= l Vay, phuong trinh da cho c6 hai nghiem la x = - = Dieu kien: X > Phuang trinh da cho tuomg duong vai x-l + ^(x-l)' *• i -2^/]r^ = ( V ^ + l)%(V]r^ + l ) ' - ( V I ^ + l) Xet ham so/(O = + Ta CO f\t) = 3t^ +2t-l; 2? (1) vai r > f\t) > vai moi f > Do ham / ( / ) d6ng biln tren [l; + oo) Vi vai X > thi ^x-\a V x - S +1 cungthuoc [l; + oo) nen (l)<»/(^/x^) = / ( V ^ + l ) ^ ^ / ^ = V ^ + l Dat u = yJx-\, , , (2) phuomg trinh (2) tra w - = yju^ -1 <^u^-u^ +2u-^ = 0^(u-2){u^ + w + 4) = < ^ u - < r > x = Vay nghiem cua phucmg trinh la x = Dieu kien: x > - Xet hai truong hop sau TH J X > Khi x^ - 3x = x(x^ - 4) + x > x > Vx + Phucmg trinh khong CO nghiem truong hop r//2 - < X < Dat X = 2cos/,0 < ^ < ;r m Khi phuong trinh da cho tra 8cos^r-6cosr = yJ2(\ cost) <=>cos3? = c o s 3t = - + k27r 3t = - - + k27r^ Do ^<t<n kAn t= t= k27r ATT nentac6 U = —,tj = An— • 5^ ATT ATT Suy nghiem cua phuong trinh da cho jc = c o s — , x = cos — + X- 2>0 Dieu kien: - x ^ x^+x-2-x#0 x<-2 x>l,x:7t-,x;^2 Bat phuong trinh da cho tuong duon voi <-x-^l x^+x-2 > (1) +X-2-X ( - x ) x^ Xet hai truong hop sau: a) x < - K h i d - x - V x ^ + x - > 0,(Vx^ + x - - x)(3 - 2x) > nen(l) thoa man b) x>l,x ^—,x ^2 Lap bang xet dau vl trai cua 11 (1) theo dau cua thua so — <x<2 cua no ta dugc nghiem truong hop x>2 "x<-2 Vay, bat phuong trinh c6 nghiem 11 — < x < x>2 180 Dieu kien: x>0,xi^\ Khi hit phuong trinh da cho <=> +) < X < 1, ta CO (1) « V2x + - • x-\ x-\ (1) < V3X + 25 <=> V2JC.VXTI<12 +x-72<0 <::> -9 < X < Suy < X < la nghiem + ) x > l , t a c (1) « V2JC+VX + >V3x + 25 « x < - v x > =^x>8 Vay bat phuong trinh da cho c6 nghiem: < x < 1; x > Dieu kien: x > - Bat phuong trinh da cho tuong duong voi x^ - 3x + 2(x - 2)Vx + l < o > ( x - + Vx + l ) % Tir giai dugc nghiem - < x < 10 Phuong trinh da cho tuong duong voi ^fl^ + \ x «(x-l)x(x-2)-0 < » x = 0,x = l , x = 181 ChUcfng PHtfCfNG TRINH, BAT PHlJdNG TRINH MU A Tom tit ly thuylt Cac cong thijfc bien doi dai so: Duai day la cac cong thiic h'lkn d6i ca ban \h luy thua, ap dung de bien doi cac phuang trinh mu: , in oam,^ < a a' a a'"-b'"=iabr,^ = vai a , > , m E ' I— a"* = _ , ^ a ' " = a " vai a > 0,6 e ! , / « , « e Z \ a Cac npi dung ve giai tick: • • - Tap xac dinh va tap gia tri: Ham so luy thua y = x'' c6 tap xac dinh nhu sau: Neu a e thi x € M, neu a e Z " u { } thi X G ] R \ { } , ngu a Z thi x e ,^ vox Q < a ^ \i tap xac dinh la R va tap gia tri la M"" Dao ham cua cac ham s6 (x") = x""',{a'')' = a'\na • Ham s6 mvi y ^ - V a i a>\, - Giai han cua ham so: • N6u < a < thi cac ham s6 y = a'' nghich bi6n - Neu a > thi cac ham SO - Tinh dan dieu cua ham s6: • , = a"" dong bien tac6 cac giai han sau lim a" = +oo, lim a" = JC->+<» - X->-00 V o i < a < ta CO cac giai han sau lim a" = 0, lim a"" = +oo 182 Cdc dgingphirfftig trlnh ca ban: 5^,;; ; -n, , ? / :i) • Phuongtrinh a" =m vai ni>0, luonc6nghiemduy nhat la x = log„m • Phuong trinh â^"^ = ấ-"^ <=> / ( x ) = g{x) tinh don dieu cua ham s6 miị • Phuong trinh a^*""' =ft^*""^voi a^b thi tinh logarit ca s6 a true tiSp hai ve, ta duge - ~ log4a^^^') = log4M<^>)«/(x) = g(x)-Iog,6 - Day la cae dang eo ban va noi ehung hau het cae phuong trinh mu va logarit tir ea ban den nang cao deu phai thong qua Trong goc nhin nay, eon mot so dang bien nhu phat hien tinh doi xung de chia va dat §n phu, biSn doi ap dung eong thuc thich hop de dua ve eung ca so, Cdc dfing bat phuang trlnh mu ca ban: Tuang ung voi cae dang cua phuong trinh, bat phuong trinh ciing eo eac dang cabannhusau: • Dang a'' > vai < o 7i : ta c6 truang hop nhu sau: " - N6u < thi bit phuong trinh nghiem dung vai moi x - Ngu > thi ta CO x> log„h neu a > va A:< log„h neu a<\ • Dang a" <h \(A ^ <a^\ ta eo truang hop nhu sau: - Neu * < thi bat phuong trinh v6 nghiem - NSu i > thi ta CO jc> log^ h nlu o < va x < log„ h neu a > • Dang â*''^ > ấ^''^ vai < a ' thi f(x) > g{x) ngu o > va / ( ^ ) < ^(^) rieu < o < - Tuy nhien, ro rang vice giai cae bat phuong trinh thi doi danh gia true tidp la khong kha thi va khong phai la mot dang thuc nen viec bien doi cung khong the linh boat duge Trong phan nay, ehung ta se nhac lai mot dinh w * w * li lien quan den viec xet dau cua ham so tren mot mien lien tue nhu sau: Cho ham s6 f (x) lien tuc tren mien D Gid sir phuomg trinh / ( x ) = c6 dung n nghiem phan biet la x^,x^,x^, ,x^^ D Khi do, giira cdc khodng 183 = 1,«-1 thi ddu cua ham so f{x) khong doi vd triing vai ddu cua mot diem bdt ki thuoc khodng Ta ihky rSng k^t qua cung tuong tu nhu viec lap bang xet diu cho mot ham so, mot cong viec kha quen thuoc Khi do, nlu xet mot bit phuomg trinh CO dang / ( x ) < hoac / ( x ) > , ta chi can xet phuong trinh tuong ung la / ( x ) = roi giai no de tim cac nghiem, xet ddu cua ham s6 giua cac khoang (thay cac gia tri cu thI vao) r6i so sanh voi yeu cSu 6k bai la hoan tit Cach giup ta tan dung dugc cac bien doi dudi dang dang thuc cua mot phuong trinh cung nhu han che dugc viec lam dung cac tinh chat cua ham s6 va tinh toan phuc tap Ngoai ra, ta ciing c6 cac danh gia sau (dang rut ggn cua cac danh gia da neu d tren): Voi < a,6 ?t va jc e R thi a"-b" cung diu voi - Voi Q<a^ \a x,;;e]R thi a^-a^ cung dau voi - (a-\)(x-y) x{a-b) B PhLrcyng phap giai 1) Phuang phap bien doi ve cung ca so hoac lay logarit hai ve Nhan xet: Phucmg phdp da duac giai thieu noi dung ly thuyet a tren O day, ta se sir dung thudn cdc bien doi dai so lien quan cung nhu tinh chdt ca bdn cua hai ham nguac la f (x) = a" vd g{x) = log^ x thong qua dang thuc f{g{x)) = gif{x)) = x Vi du Giai phuong trinh 16 = v -1024^ 2112 Lai giai *) Dieu kien xac dinh x > 0, x /3x+2> 12X+8 J = 2"3 2'°^ « l Phuong trinh da cho tuong duong voi n = 2>°^ « 2"^^3 , o ^ ^ + ^ = 1OV^ x-1 Khu mau va nit ggn, ta dugc 4x + = 3(x - l)\/x 184 Dat/ = 0,taCO 't = 3/^+2/ + l = ' De thay phiromg trinh thu hai v6 nghiem nen phuang trinh tren c6 nghiem la t = Vai t = thi X = nen phuang trinh tren c6 nghiem la x = Vi du Giai phuang trinh 2"' = ' ' L&igidL Ta c6 4-2' 2'*' = 16'' <=> 2'" = T'' _ ^2-" -.2"^ « 2'" = ' " « •>4x 2'^ = T2X + <=>4x = x + 2<=>jc = - Vay phuang trinh da cho c6 nghiem nhat la ^ = " j • 2x-3 V i du Giai phuang trinh 3^"' • ^ = 18 LoigidL *) Dieu kien xac dinh jc 9^ o Ta c6 2x-3 3^^-'-4^ 4x-6 =18«3^'-'-2^ 6-3x = - ' « ^ ' - " =2" Lay logarit ca so hai ve, ta dugc x'-4 , = 6-3x log, « ( x - 2)(x + 2)x + 3(x - 2) logj = + 2) + log3 2] = (x - 2) x-2 x=2 =0 x(x + 2) + 31og32 = (x + 1)' +31og3 - l = D I thiy phuang trinh thu hai v6 nghiem nen phuang trinh ban dau c6 nghiem la X = V i du Giai phuang trinh sau 2^"' • 3" = 4^' • ^ L&igidL *) Dieu kien xac dinh: x^-l 185 2x Taco 2^^'-3'=4^ • 36^^' o • 6" = ' •6^^'o4'-^ =6'^' Lay logarit ca so hai v l , ta duQfc x+1 "l-x = , log4 « (1 - x) [(x +1) - X log4 = <=> x=l (x +1)' - x l o g ^ = " [ x ' + x(2 -log4 6) + = D I th4y phuomg trinh thu hai v6 nghiem nen phuomg trinh ban dhu c6 nghiem la x = l V i du Giai phuong trinh sau ( ^ " ' + ( s ^ = [l"''- • (2"* J Z,^i^/fl£ Phuomg trinh da cho tuomg duomg voi O 26 • 5^ =26-r « 5' =r o (x' + 3x) log^ = x' + 6x « x[(x + 3) log2 - (x + 6)] = "x = 125 logjS-l 64 6-31og,5 , ' i^^ - • Vay phuomg trinh da cho c6 hai nghiem phan biet la x = 0,x = log 64 125 V i du Giai bdt phuomg trinh sau 3^^''^' > (DH Bach khoa Ha Noi 1997) L&i gidL * ) DiSu kien xac dinh: x^ - 2x > <» Ta CO 3^^ > / I X > J- , x<0' !' V (*) -X Neu | x - l | - x < < : > | x - l | < x < = > - x < x - l < luon dung ^ , > - thi bat phuomg trinh (*) _ 186 NSu \x-\\-x>0^x<- thi M t phuong trinh (*) tra ylx^-2x binh phuong vS, ta dugc - 2x > - 4x + <=> >\-2x, - 2x +1 < 0, v6 nghiem Do do, nghiem cua bat phuong trinh da cho la x > : 4" +2x-4 Vf du Giai bat phuong trinh sau < x-1 (DH Van Hoa Ha Noi 1997) Led giai *) Dieu kien xac dinh x*\ Ta c6 x-\ Ta CO truong hcfp: - Neu -Neu < 4^-2<0 4^<2 x<log,2 = x-l>0 x>l x>l 4^-2>0 f4^>2 x-l<0 [x<l , khong thoa man x>log42 = o-<x<l x<l Vay nghiem cua bat phuong trinh da cho la x e V ?5X L\ Bai tap phan 1 Giai phuong trinh sau 5^+5^*' +5^"' = 3^+3^^' +3^"' Giai phuong trinh sau 5' • ^ = 500 (DHKinh ti quSc dan 1998) Giai phuong trinh sau 5"^ - + 5^"'-125 = 4-5 Giai phuong trinh sau (41 + 29^12)"^ = (V2 -1) ,sin4j: Giai phuong trinh sau (10 + 6V3)'''"^ = V(V3+1) Giai b i t phuong trinh sau 343 • 1"'''^ > 2,2-1 Giai b i t phuong trinh sau ————— < (DH SP Ha Noi 2001) 187 Huong din bai tap phan 1 Xet phuomg trinh 5" + 5""' + 5^"' = 3" + 3""' + 3""' Ta c6 5^(1 + + 5') = 3^(1 + + 3') « 5^-31 = 3^-31 «5^=3^« .3j = < » x = Vay phuong trinh da cho c6 nghiem la x = Xet phuomg trinh 5" • ^ = 500 D i l u kien x ^ Ta CO 3(x-l) 5^-2 ' 2- 3x-3 =2'-5'<=>5^-'=2'" - 3-x «5^-'=2- Lay logarit co so hai vS, ta c6 (X-3) log, = 3-x o(x-3) log2 + - = 0<::> y X X = x = -log5 xj Vay phuomg trinh da cho c6 nghiem la x = hoac jc = - log, Xet phuomg trinh sau Is"-ll + ls"^^-125 =4-5'*\ Taco ^ - l > » x > va ^ * ^ - > « ^ > « x > Ta xet cac truomg hop sau: -NSu x<0 thi - ^ + - ^ ^ ' = - " ' « = - ' o ^ = l < : > x = 0, thoa man -Ngu x > l thi 5"-1 + 5^*'-125 = - ^ " ' « 74-5^=-126, v6 nghiem -Neu < j c < l thi ' - + - ^ ^ ' = - " ^ ' « - " = " =1 <=>x = 0, loai Vay phuomg trinh da cho c6 nghiem nhit la x = Xet phuomg trinh (41 + 2972)""' = (V2-1)"'^"\ Ta CO 41 + 29V2 = (V2 +1)', V2 - = (V2 +1)-' nen lay logarit ca s6 V2 +1 hai v8, ta dugc 188 x=l 5(jc-2) = - ( x ^ + x + ) « x ^ + x - = 0<=> x = -7 Vay phuomg trinh da cho c6 nghiem \a x = \,x = - Ta CO 10 + 6V3 = (Vs +1)^ nen phuomg trinh da cho tuomg duomg voi (73 +1)'^'"^ = (73 + if""'' « sin X = sin X • cos X • cos 2x » sin x(cos X • cos 2x - 3) = Do cos X • cos 2x - < 0, Vx nen phuomg trinh chi c6 nghiem la sinX = <=>X = kn:,keZ Xet bit phuomg trinh 243 • 2"'"' > 32 • \ Ta CO 7' • 2"'-' >2' -r <^ 1"-^ < 2"'"' Lay logarit ve theo ca so 2, ta dugc ( x - ) l o g < x ' - < » ( x - ) ( x + 3-log2 ) > <=> "x>3 X < -3 + log2 Vay nghiem cua bit phuomg trinh da cho la x G (-00; -3 + log^ 7) u (3; +00) Xet bat phuomg trinh '^^ < DiSu kien xac dinh 3"^ - 2^ ^ » x ;t ^ 2-3^-4-2^-(3^-2^) ^ 3^-3-2^ „ ^<0<=> <0 Taco : 3^-2^ 3'-r Ta xet truong hop: .3V f ' - - ^ < <=> < 3^-2" > x>0 3^-3-2^ > 3^-2^ < 2) <3 <=> < X < logj _ ^, khong thoa man x<0 Vay nghiem cua bat phuomg trinh da cho la x e 0;log3 J 189 2) Phuong phap dat an phy Nhan xet: Trong phucmg phdp nay, ta se gap cdc phucmg trinh c6 su xudt hien cua mot bieu thirc ndo nhieu Idn vd dieu ndy dinh huang cho ta dat bieu thirc 1dm an phu de bdi todn duac dan gidn di Chit y rdng dat dn phu nhu the, cdn tim mien gid tri cua bieu thirc de tuang irng suy dieu kien xdc dinh ciia phuang trinh vai biin mai Trong phdn ndy, ta c6 cdc dang dn phu chu yeu nhu: • /(gix)) dat t = = thi dat t = g{x), dua vi phuang trinh f (t) = , dua • A • fl'^+ B • At^ -\- Bt + C = 0, a^b +C • ' = 0, chia vi cho a'«'^* > ',duavg ^r^+5/ + C = Vi du Giai phucmg trinh 27^ = ^ + 3" - LM gial Ta bien doi nhu sau: (3' )^ = • (3' )^ + 3" - Dat ^ = 3"^ > thi = 3/' + / - « - 3)(/ -1)(/ +1) = 3^=1 t =\ >=3 /=3 'x = \ x =0 Vay phuofng trinh da cho c6 nghiem la x = 0, x = x-^ jc-3 3x+l Vi dy Giai phuomg trinh 8^*^ + 2^^^^ = _ L^/^/ai *) Dieu kien jf i - Ta CO - ^ ^+2+2 ^"^2 o4 x+2 <»8 ^ ^2 ' _2 + 2-2 _2 =8-2 -3-2 -2 = _ i 190 Dat y^l -— > o , t a c / - ; ; =-1«(:i; + l)(2>'-l)'= = - ^ -1 o = - l « x = da xcho + 2c6 nghiem la jc = Vay phuomg trinh Do Vi du Tim m dS phuomg trinh sau c6 hai nghiem trai diu: (/M + 3)16"+(2w-l)4'^+7M + l - / Lei gidl Dat / = 4"^ > Ta c6 phuomg trinh {m + 3)/^ + {2m -1)/ + w +1 = Ta can tim m cho phuomg trinh c6 nghiem t^,t^ thoa man < /, < < ^2 Tmoc het, ta can tim dieu kien de phuomg trinh c6 nghiem deu duomg, tucla (2m-l)^-4(w + l)(m + 3)>0 A>0 11_ S > <=>-i 2m-\ m + 7>>0 20 P>Q m+1 >0 l/n + Dat tilp M = r -1 <=> / = M +1 thi ta dua \k phuong trinh (m + 3)(« +1)^ + {2m - 1)(M + l) + m + l = 0<»(w + 3)M^ + (4m + 5)M + 4/w + = Tiep theo, ta can tim m cho phuomg trinh tren c6 nghiem u trai diu, dieu tuomg duomg vai ^ , (w + 3)(4w + 3)<0«-3</n<— ' ' Ket hcfp cac dieu kien lai, ta c6 -1 < w < - — Vi du Tim m dl phuomg trinh sau c6 nghiem nhat thuoc [0;1] 22-2^ _ +m+2=0 191 L&igidL Phuang trinh da cho tuong duong vol ( 2X 'K) ^x-\ + m + = Dat / = — > , ta thay xe[0;1]<=>/e[1;2] Ta can tim m cho phuong trinh da cho c6 dung nghiem thuoc midn [1;2]: -t + m + = Dat / ( - r - / + vol ^ e M , t a c / ' ( = / - l hay/'(O = o r = | Ngoai ra, /(1) = , / ( ) = nen lap bang bifin thien, d l dang suy / ( = -m CO nghiem nhit thuoc [1;2] -4<m< -2 Vay dieu kien can tim cua m la -4 < m < - Vi du Giai phuomg trinh ^ ^ ' + 2• ^ " =2-4" L&igidL Phuomg trinh da cho tuong ducmg voi 2^^^^^ + • 2^"^'^" - • 2'" - o • 2' Dat / = 2^'^'-" > , ta dugc r + / - = + 2.2-"^'-' - = / ' = "/ = - « • t= Ta chi nhan nghiem r = ^ va 2^'-' =-<^l[I+5-x = -\<:>l!7T5-x = -\<^x + = {x-Xf < » x ^ - x ^ + x - = « ( x - ) ( x ^ + ) = 0<»;c = Vay phuomg trinh da cho c6 nghiem la x = V i du Giai bat phuomg trinh 36^'^""'+64"'^"''> 100-48"'^"-* L&i gidL Bat phuomg trinh da cho tuong duong voi 36 • ( ^ ' ^ ^ - ' + 64 • (8^'"^-')'> 100 • 6^'""-'• 8^'^^-' / «36 -100 _ \ +J:-6 + 64 > v4 192 « -25- + 16>0 4) Dat / = >0 thi 9/^-25/ + > < = > ( / - l ) ( / - ) > < » 4, - Vai r < l , t a c x^+x-6>0<:> x>2 16 /<1 t> '/'^-^ 16 -Vai / > —,tac6 v4j -i-Vi7 ^ <^x^+x-6<-2<;^x^+x-4<0<^ 4j Ket hop lai, ta dugc x e (-oo; -3 -i-7i7 V i du Giai phuang trinh (2 + -i+Vn u[2;+oo) + (2 - V3) ^2-V^L&igidL Phuang trinh da cho tuong duang vai (2-fV^P « (2 + Dat ^ = (2 + V N/3 ( - V ) + (2-V3p"-'(2-V^) = + (2 - V = > thi = (2 - > / v a ta dugc t + - = 4<=>t^-4t + \ 0^ t - V o i / = 2-73 « ( + >/3r'-'^ =2-73 » x ' - x = - l « x = l - V a i t = + S<^{2^SY-^' =2 + S<^x^-2x = \<^x = \±y[2 Vay phuong trinh da cho c6 nghiem la x = 1, x = ± V2 -1 + Bai tap phan Giai phuomg trinh 9^'"'"' -10- 3^'^^"' + = Giai phuomg trinh 2' _22+ • =3 (Dy bi DH khSi B 2006) (DH khoi D 2003) (DHSP Ha Noi 2000) Giai bat phuong trinh 3'' - • 3"^"^ - • ^ > (DH Y Ha Noi 2000) Giai phucmg trinh 2'^-6-2^-^3i^ + p = l Giai bit phuomg trinh 27'' +12"^ > • 8"^ Giai phuomg trinh ^Jl + ^l-l^" = (l + 2^1-2^^)• 2" Giai bat phuong trinh V9 + - ^ - ^ + ^ > (DH SP Ha Noi 2012) Giai phuomg trinh ( - + (6 + s[\\y zx-2 =9A-5' Giai bit phuomg trinh Vl5-2''^'+l > 2^^ -1 + 2"^' (DubikhdiA2003) Huong dan giai bai tap phan Xet phuong trinh 9'''^"-' -10-3"'^"-' +1=0 Dat / = 3''""^ > thi phuong trinh da cho viSt lai 't = \ r - / + l = 0<=> -Vdd / = thi ^ ' ^ ' ' - ' = l « x ' + x - = o • x=l x = -2" x=0 ^ - V o i t = - thi ' ' " ^ - ' = - « x ' + x = « 9 x =- l • Vay phuomg trinh da cho c6 nghiem la x = -2, x = -1, x = 0, x = Xet phuomg trinh ' ' - 2'^""^' = Dat / = 2'' , ^, v >0,tac6 / - y = < » / ^ - / - = « '••:w ; , ^V + £ - ^ /=4 Ta chi nhan nghiem t = 4>0 va 2'' x=2 = <=> x" - x = <=> x=- l Vay phuong trinh da cho c6 nghiem la x = - l , x = Xet phuang trinh 2'"-6-T + ^ = 2^ Dat / = 2">0 taco ^ _ / - l + l ^ = l « / ^ - / ^ - + 3-/- V =1 t 2^ t — = = l<=>r^-/-2 = 0-» Ta chi nhan nghiem t = 2>0 va x = log^ r =- l / =2 Vay phuang trinh da cho c6 nghiem nhit la x = Xet b i t phuang trinh 3'" - • " ^ ^ - • ^ > Dieu kien xac dinh: x > -4 Bat phucmg trinh da cho tuang duang voi > « 32 2^/^ _ - - V ^ - > 3^' - - ' 2^-9-3^^ Dat / = " - ^ > thi - / - > <=> (/ + l ) ( / - ) > » / > - Vai / >9<=>x-Vx + >logj9 = o x - > Vx + Dat dieu kien x > 2, binh phucmg ta dugc x^-4x + > x + 4<=>x^-5x>0<=>x>5 Ket hap lai, ta thay nghiem cua bat phuang trinh da cho la x e (5;+QO) Xet bat phuang trinh 27^ +12^ > • \ Ta bien doi nhu sau Dat / = / f v2 > thi ta CO >2 2) + / > <^ (/ - l){t^ + / + 2) > <=> / > v2y Dodo >l«x>0 195 Vay nghiem cua bat phuang trinh da cho la jc e (0; + 0 ) Xet phuang trinh + = (l + 2^1-2'^) • 2' DiSu kien - 2^"^ > <:> X < Dat t = yjl-l^' thi < / < 1, suy 2^' =\-t^ <^2' = yjx-t^ nen phuang trinh da cho tuang duang vai r=0 « = (1 + 2tf (1 - «t{At^ - 3) = « - V a i / = 0, tadugc V l - ' " = o ' ' = l o x = -Vai = - , t a d u a c 2"" =\ = -<^2x • 4 = -2<^x = -\ Vay phuang trinh da cho c6 nghiem la x = 0, x = - Xet b i t phuang trinh V9 + - ^ - ^ + ^ > x<A Dieu kien • x<4 4-x>0 + - ^ - ^ >0 (3^-9)(3^+l)<0^|3^<3 <:>0<X<4 Dat ^ = ^ > t h i t a c o A/9 + 8/-?' > « ^ + 8/-^' > - / (*) - Neu / > <=> - / < thi (*) nghiem dung - N8u / < <» - / > thi (*) tuang duang vai + / - / ' > (5-0^ « + / - / ' > 25-10/+ < » / ' - / + 8< <:>!</<8 Trong truang hap nay, ta dugc < / < Ket hop lai, ta c6 nghiem la / > hay ^ > o V4-Jc > x<4 Vay nghiem cua b i t phuang trinh da cho la x e [O; ) 6-vriY r6+vrT Ta bien doi nhu sau A 94 25 196 Chu y r&ig (6 + y/U 6-vrT = nen neu dat / = 6+ViT >0 thi \ '6-VrT t Ta CO phuang trinh r + - = — « 25?^-94t + 25 = 0^t = "^"^ -^^VlT 25 t 25 ^ 47+i2Vn ^ Vai r = , , ta C O 25 ' (6+VrT 47 + 12^/^T <»x 25 = [ Vai = ^ Z ^ , t a c ^ ^ ^ ^ 25 47-12Vri 25 <» X= Vay phuang trinh da cho c6 nghiem \a x = ±2 Dat / = 2^ >0,tac6 V30/ + > | / - l | + 2/ -2 ' - - - Ta CO truang hap sau: - N g u / > l , t a c V307+T>3f-1 Chiiy rang / - l > vai / > ! nentaco ^ferf 30t + \>i3t-\f ^9t^-36t<0<^0<t<4 Do < 2^ < <:> < X < - N8u < / < 1, ta CO V307TT > f +1 Chu y rang ^ +1 > nen ta c6 30/ + l>(? + l ) ' « / ' - ^ < « < / < Do do, truang hop nay, ta c6 / < <» 2"" < « x < j - \• Ket hap lai, ta duac x < la nghiem cua bat phuang trinh da cho 3) Phuang phap phan tich Nhan xet: Trong nhieu tinh huong, ta gap phdi cdc phuang trinh khd rdc roi vd ddi hoi cdn phdi phdn tich bieu thirc tuomg ung thdnh nhdn tic de Idm dan gidn hoa phuang trinh han roi xu ly rieng le timg phuang trinh nhdn tu Ta Cling CO the coi x hogc bieu thuc ham mu lien quan den x Id bien, thdnh phdn lai Id tham sd de tinh delta vd de bien ddi han 197 V i du Giai phuong trinh 25 • 2^ -10^ + 5^ = 25 L&igidL TabiSn d6i nhu sau 25-2^-2^-5'+5"-25 = « 2^(25 - 5^) + (5^ - 25) = » (2' -1)(5^ - 25) = 2^ = <=> y = 25 x =0 x=2 Vay phuong trinh da cho c6 nghiem la jc = 0, x = V i du Giai phuong trinh sau 4x' + 2x.2''^' +3.2"' = x'.V' + 8x +12 Lcfi gidu Phuong trinh da cho tuong duong voi x'(2"'-4) + 2JC(4-2"') + 3(4-2"') = • (2"' - 4)(x' - 2x - 3) = <=> 2"' - = V - 2x - = <=>x = ± \ ^ , x = - l , x = Vay phuong trinh da cho c6 cac nghiem la x = ±V2, x = -1, x = V i du Giai phuong trinh ( V - l ) " " ' % x ( V = x ' +1 L&igidL f r- - *) Dieu kienxac dinh: X > Phuong trinh da cho tuong duong voi (nhan them v6 cho (yfs+lf'^''' > ) : (V^+if"[(V3-i)'""+x(V^+if'^] \ - = (V^+i)'"^^(x^+i) «2'-'+x(V3+lp"=(V^-flp(x^+l) O x + x(V^ + l p " " ; =(V5 + l ) ' " " ( x ^ + l ) s / 1— \21og2J: 21og2Jt + = V3 + <=> X (x^+l)o = -^^ Ta thay dang thuc o a b + b = ab + a {a-b)(ab-1) = <=> 'a = b ab = \ a 198 Ta CO hai tnrcmg hop sau .-N ; - - N6u X = (V3 +1)'°^^'' thi lay logarit co s6 hai v l , ta dugc logj jc = log2(>/3 + l).Iog2 X <=> logj X = <=> X = 1, thoa man dieu kien xac dinh - Neu x(\f3 +1)^°^'" = thi cung lay logarit co s6 hai vg, ta duac log2X + log2xlog2(>/3 + l ) = < » l o g X l + log2(>/3+l) <=> log2 X = «> X = =0 ma'! Vay phuang trinh da cho co nghiem nhat la x = Vi du Giai phuang trinh 4" - 2"^' + 2(2" -1) sin(2" + -1) + = (Du bi khSi D 2006) L&igidL Phuang trinh da cho tuang duomg voi (2'" - 2.2" +1) + 2(2" -1) sin(2" + -1) +1 = o (2" - ) ' + 2(2" -1) sin(2" + > ' - ! ) + sin' (2" + 2" - + sin(2" +y-1)J' <=> -l2 -1) + cos' (2" + -1) = + cos'(2" + - ) = 2"+sin(2"+>'-l) = l _ cos(2"+>'-l) = Do cos(2" + -1) = sin(2" + -1) ± Ta CO hai truong hop sau -Ndu sin(2" + >; -1) - thi 2" = 0, v6 nghiem -N6u sin(2"+>;-l) = - l thi 2" = < : ^ x = l Suy sin(j + l) = • " TT = -—-\ k27r,k e Z Vay phuang trinh da cho c6 nghiem la x = l,>' = - — - + kin,k e Vi du Giai bat phuang trinh 2"^'+ (5x'+11) • 2'-" - x ' < 24 - X[1 - ( x ' - 9) • 2^"" LcigiaL Dat r = 2",r > 0, bdt phuang trinh da cho tra thanh: , 10x'+22 2t + ^/ ^'-9x x'<24-x + 199 « / ^ - [ x ^ - x + 24)t + 22 + 9x + \0x^-x^ I <0 Taxem 2/^ -[x^ -x + 24)t+ 22 + 9x + l0x^ -x^ =0 laphuomg trinh i n / Tac6 A = (x^-x + 24f - ( 2 + 9x + lOx' - x ^ ) = [x' + 3xf - 40 ( x ' + 3x) + 400 = ( x ' + 3x - ) ' Suy ra, phuomg trinh c6 hai nghiem phan biet: t = -x^ + - X 2 / = -x + ll , +1 Do do, bat phuong trinh da cho tuong duong: (2/-x^-x-2)(/ + x-ll)<0 2-2^-x'-x-2>0 2r-x'-x-2>0 <=> r+ x-ll<0 2'+x-ll<0 <=> 2-2^-x'-x-2<0' 2/-x'-x-2<0 r+ x-ll>0 Lb' ' + x - l l > Xet ham s6 / ( x ) = 2" + x -11 thi / ' ( x ) = 2Mn +1 > nen no dong biSn tren R Mat khac, / ( ) = nen ta c6 / ( x ) > « x > va / ( x ) < » x < Xet ham s6 g(x) = - " - x ' - x - tren M Taco g\x) = • 2^ • In - 2x ii> g'Cx) = 2-2^ (In 2f-2=^ g"'(x) = • 2^ (In 2)' > Mat khac g(0) = g(l) = g(2) = nen phuomg trinh g(x) = c6 ba nghiem la X = 0,x = l,x = Lap bang xet ddu ciia ham s6 g(x) tren cac khoang (-oo;0),(0;l);(l;2);(2;+oo), ta c6: • g(x)>0«xe(0;l)u(2;+oo) g(x)<0«XG(-oo;0)u(l;2) Do do, bdt phuomg trinh da cho tucmg ducmg vai: 200 \g(x)>0 \fix)<0 0<x<lvx>2 X < <=> 0<x<l g(x)<0 x<0vl<x<2 f(x)>0 x>3 2<x<3' Vay tap nghiem cua b i t phucmg trinh da cho la (0;l)u(2;3) Vi du Cho bdt phuong trinh xyllx-x^ < - ax2" + al^yjlx-x", tham s6 voi a la - r V ' • a) Giai bat phuong trinh a = - ^ .^ b) Tim a de bat phuong trinh c6 nghiem x > LoigidL a) *) Qihu kien 2x - x' > « < x < Voi a = - l , taco xV2x - x' < x' + x2' - 2" V2x - x' S-^^CH^ » V x - x ' ( x + 2^)<x(x + « ( x + 2")(V2x-x' - x ] < Chiiy rang x + 2'^ >0 nen yjlx-x^ <x<ei>2x-x^<x^<::>x-x^<0<=>0<x<l Vay nghiem cua b i t phuong trinh la x e (0; 1) v^ b) Ta bien doi nhu sau: xV2x - x^ < x^ - 0x2"" + al^-Jlx ulM - - x^ <:>x(V2x-x^ - x j < a ^ (V2x-x^ - x j « ( V x - x ^ - x j ( x - a ' ^ ) < Do X > nen x-x^ <0<=> 2x-x^ <x^ <=> yjlx-x^ x-al' >0^a< - x < , suy X Xethams6 / ( x ) = — , x e [ ; ] thi / ' ( x ) = 2^-x2Mn2 , - va / ' ( x ) = <z> " - x M n = « X = — e [1;2] In2 ^ ^ Khao sat ham s6 / ( x ) tren mien [1;2], ta thdy / ( x ) > /(1) = ^ va suy a < ^ 201 V i du Giai phucmg trinh 16' + 48^ + 96^ + IT + r = 18" + 64^ + 24" +54^+108' LMgidL Dat ^ = 3"^ > 0, ta CO - , 2'" + ' ' / + ' ' / + / ' + / ' = ' / ' + * ' + 2'"r + ' / ' + ' ' / \ Ro rang nSu thay t = T thi c6 ding thiic nen phuong trinh tren c6 nghiem / = 2"^ Taphan tich nhu sau : • - -^, • (2'" - ' " / ) + (2'"/-2'"e) + ( ' " t - ' " ) + [t' -2U^)-^[t' » 2'" ( ' -t) + 2''t(2' -1) (2" +t)- 2'" (2" -t)-t' <^[2''-t)(2''+2'U o (2" -1) [[2'^ o (2" - ] [2'' - + 2^U^-2''-t^-t') +1[2"' -t')- - -2U') = [2' -t)-^ (T -t) = =0 2'^ [2'' ' =0 mmA -^s ) ( l + / - 2'") = o ( ' - 3") (8' - 9") ( l + 3" - ' ) = Ta CO cac truong hop: ^ * -Neu ^ - ' = o ^2Y -Neu 8"-9'=0<=> - Neu l + ' - " = o = <=>x = v4y 4; = Ve trai giam theo jc nen phuong trinh CO khong qua nghiem Ta lai thay x = l thoa man nen phuong trinh c6 nghiem nhdt la x = Vay phuong trinh da cho c6 nghiem la JC = 0, x = *" H « ^' i B a i tap p h a n Giai phuong trinh 4x^ + x • 3' + 3'"' = 2x' • 3' + 2x + Giai phuong trinh 52-^-3^-2+5^^+3x^2 ^ 625 125"' +1 Giai phuong trinh • 3" + • 2" = 24 + 6" Giai phuong trinh (DH QuSc gia Ha Noi 2000) • T +r{2-7>x) = 6x^-x'-\\x +6 202 Giai bat phuang trinh 4x'+x-2"'^' + 3• 2"" > \log,4r / — • + 8x +12 > logj-t Ino r Giai bat phuang trinh (^S + dY'" - ( V - ) T2 C Tim a cho phuang trinh sau c6 dung nghiem ^ • 3^ + o x ' + • 3^ + a x ' = X • 3^"'+ 3a • Jc' Huang dan giai bai tap phan l.Xet phuang trinh x ' + x• 3^ + 3^"' = x ' - " + x + Ta bien doi nhu sau x ' ( - - ^ ) + x ( ^ - ) + 3^"'-6 = o - x ' (3" - 2) + x(3^ - 2) + 3(3^ - 2) = <=> (3^ - 2)(2x' - x - 3) = 3^ = 2x'-x-3 = X = logj <=> , x = -l,x = - Vay phuang trinh da cho c6 nghiem la x = logj 2, x = - , x = — Xet phuang trinh = 625 • 125" + Ta b i l n d6i nhu sau -2x'-3x+2 2x'-3x + = y+3x +2= -1 = Phuang trinh v6 nghiem x = - l , x = -2 Vay phuang trinh da cho c6 nghiem la x = -2,x = - r* Xet phuang trinh • 3"^ + • 2"^ = 24 + \a bi6n d6i nhu sau 8(3" - ) + 2^(3 - 3") = « (3" - 3)(2" - 8) = » 3" = x=l 2" = x=3 Vay phuang trinh da cho c6 nghiem la x = l,x = Xet phuang trinh x' • 2" + 2" (2 - 3x) = 6x' - x' - Ix + 203 Ta bien doi nhu sau 2" (jc' - 3x + 2) = -{x - \){x - 2){x - 3) « 2^(x - l)(x - 2) + (x - - 2)(x - 3) = X ^ ( x - \)ix - 2)(2" + X - 3) = « = l,x = 2'^ + X - = Ptiuomg trinh cuoi c6 ve trai d6ng biSn nen c6 khong qua nghiem Ta nhdm thiy x = thoa man nen phuang trinh da cho c6 nghiem la x = l,x = Xet bat phuang trinh 4x' + 2x • 2"'"' + • 2"' > x' • 2"' + 8x +12 Ta bien doi nhu sau x ' (4 - 2^') + 2x(2^' - 4) + 3(2'' - 4) > <:> (T' - 4)(x' - 2x - 3) < « (2^' - 4)(x + l)(x - 3) < Chii y r i n g " ' - - o x^ = » X - ±V2 Lap bang xet d i u cua b i l u thuc, ta thu dugc nghiem cua hk phuang trinh la - X G ( - > ^ ; - ) U ( V ; ) Xet b i t phuang trinh (3V5 + 6)'°''' -(3>/5 - ) ' ° " ' > x ' - Dieu kien x>0 Ta thay x = thoa man bat phuang trinh Xet x^l Ta CO (3N/5 + 6)(3V5 - 6) = 45 - 36 = Suy _ log,((3^/5+6){3^/5-6)) _ logJ(3^/5+6)+log,(3^y5-6) _ log,(375+6) Iog,(3,/5-6) Do do, ta bien doi nhu sau log,(3^/5+6) _ X logj(3V5-6) X ^ \ogy{i45+6) ^ X 1083(375+6) _ N—»^ X ' « log3(3r^-6) * X log,(3V5+6)_ log3(3v/5-6) X * X log,(3x^-6) ^ X _ ^106,(375+6) L _ ^log,(3V5-6) \ ^ l o g , ( V i - ) _ ^ (^log3(375-6) _ ^ ) ( ^ l o g ( / + ) _ ^ ^ ) ^ ^ ^ ^-06,(3^5-6) _ ^ ^ ^ Do logj ( V - ) < nen X > Vay nghiem cua b i t phuang trinh da cho la x > 204 Xet phuong trinh • 3" + ox' + • 3' + 2ax' = x • 3'"' +3a-x\ Ta phan tich nhu sau 3"(x' - 3x + 2) + ax\x^ - 3x + 2) = « ( x ' - 3x + 2)(3' + a x ' ) = « x ' - 3x + = 3^+ax' = Phuong trinh x^ - 3x + = da c6 nghiem la x = 1, x = Neu fl > thi de thay phuong trinh thuhai v6 nghiem Dat / ( x ) - " + a x ' voi a<0 D l thiy lim / ( x ) = -oo va / ( ) = 1>0 va / ( x ) lien tuc tren (-oo;0) nen x *-co phuong trinh c6 it nhat mot nghiem am, khac voi hai nghiem x = l , x = o tren; do, khong thoa man Vay dieu kien can tim la a>0 ' ' - > > -r^ ' - i i , > - t H - ^ T J H D 3.icf tiX 4i 4) Phuong phap danh gia Nhan xet: Trong mot so truang hap, ta c6 the sic dung bat dang thirc de chtrng minh phicang trinh v6 nghiem hoac c6 khong qua mot so nghiem nao Cu the la ta se chon mot dai luomg trung gian nao de lam ca sa ddnh gid, so sdnh hai ve ; V i du Giai cac phuong trinh sau a)i2 + yf3y+(2-yf3y=4\ h){2+y[3y+{2Sy=r L&igidL a) Ta bien doi phuong trinh da cho + V3 + 2-V3 = + ^/3 — V3 , < nen ve trai cua phuong trinh la ham nghich Chii y rang < 4 bien va do, no c6 khong qua nghiem Hon nua, x = thoa man nen phuong trinh da cho c6 nghiem nhat la x = 205 b) Ta bien doi phuang trinh da cho „, , , J + V3 , 2-yf3 Chu y rSng >1 > - NIU X > thi V + V3 ~ y + 3iaX A = ^ ' r , , nen ta co cac truong hop sau: > + V3Y = 1, vaJ2-S' > nen ve trai ion hon V 2-V^ - Neu x<0 thi , 2-S V = va / + V3 > nen ve trai ciing lomhom Do do, phuomg trinh da cho v6 nghiem Vi du Giai b i t phuang trinh sau 2^"' + 3"' < 3'^"' + 2^^^' LM gidi < 3'^"' + 2'^"' Xet bat phuang trinh 2'^^ + Ta thay rSng n^u x = \i hai \h cua b i t phuang trinh bSng Ta xet cac truong hap sau -N6u X > => X + < 2x +1 thi 2"' < 2'^"',3^"' < 3'-^"' ^ T^^ + 3^"' < 2'^"' + 3'^"', thoa man _ ^ -Neu X < thi bat dang thuc a tren doi chiSu va khong thoa man 6k bai Vay b i t phuang trinh da cho c6 nghiem la ;c > r^"' + 2^'"'' = 2*-''"' + 2'"'*-' V i du Giai phuang trinh „ , i-:,., (MSG TP Ha Noi 2005) LotigidL Theo bat dang thuc Cauchy cho s6 duang thi: -3 9-4 + UJ>2V2^'^^ -2 ^^^=2-V2^'^ ^"^ >2V2^=4 Ta CO 3=°^'' + 2'"'^ < 3'°^'^ + 3""'' Ta se chung minh rSng s^"^'"^ +3™'^ < That vay, dat / - 3'°^'" Do < cos^ x < nen < r < B i t ding thuc c i n chung minh tuong duang v a i / + - < <» ( / - ! ) ( / - ) < , dung 206 / „ ^2 X -3 = 9-4 / X N / \ X =2 DSng thurc xay sin^ x = \ cos^ X = V cos^ x = <:> x = 27r Vay phiromg trinh da cho c6 nghiem nhit la x = ITT Vi du Giai phuorng trinh LcfigiaL Phucmg trinh da cho vilt lai la (2 + V2r'"-(2 + V2)' 4i 1+ -{2-42T'^' (*) Ta xet cac truong hop sau: -N6u cos2A:>0<::>cos^x>sin^x, + V2 >1 nen (2 + V r ' ^ - ( + V2n'^ <0, do 1+—>1>2-V2 nen -\cos2jr A life -(2-V2r^'^>0 Trong tnrong hop nay, phuang trinh (*) a tren khong dugc thoa man - Neu cos 2JC < 0, lap luan tuang tu truomg hgp tren ' ^ - N8u COS2JC = thi (*) dugc thoa man va phuang trinh da cho c6 nghiem la x ^ - + k-,k&Z ••=• ' Vi du Giai bdt phuang trinh sau 5^ + 4' < - ( ' + 2' +1) Ldi giai Ta thay rang J: = la nghiem cua phuang trinh 5^+4^=^(3^+2^+1) 207 Bat phiromg trinh da cho c6 thS viet lai la v4y v4 + V4y + Ta xet cac tnidng hop sau - Neu X < thi 4j ^1 V , , , +1 < — +1 = — va 4 v4y 4 > — + — + - = - v4 Suy ^5V + v4 v4 V4y ^2Y \ /1 f 2Y + < — : — = — , thoa man de bai 4'2 v4 - Neu jc> thi de dang thay rang bat dang thurc tren doi chieu va khong dung voi dh bai Vay nghiem ciia bat phuong trinh da cho la x e ( - o o ; l ) Vi du Giai phuong trinh sau 3^' + 3""^' + 3"' - 3"^'"' L&igidL Xet phuong trinh 3'' + 3^'^' + 3'^' = 3""^'^' Theo bat dang thuc Cauchy, ta c6 ' ^ 332.^ ^ 3.28 > 3 / 7 ^ ^ ^ 7 ^ ^ ' - Hon nua, ta cung c6 Tu suy 3"' + 3'''' + 3"' > 3"^'^' va dang thuc xay jc = Thu lai ta thay thoa man Vay phuong trinh da cho c6 nghiem nhat la x^2 ( Vi du Giai phuong trinh sau M( 2^+4^ V J = 144 L&igidi *) Dieu kien: x ^0 208 T ( Chu y rang vai x < thi M <4<144, dodo x > Theo bat dang thurc Cauchy ta c6 Suy ( 2^+4^ =2^+2 ^ >2V2^-2^ =2-22^^ >2-2 ^^'^ =8 va =3^+3 ^ >2V3^-3^ =2-3^^^ >2-3 ^^'^ =2-3^ =18 ^\ ( 2^ 3^+9^ >8-18 = 144 \ V ) X Dau bang phai xay ra, tuc la x = - , - = - <z> x = X X Vay phuang trinh da cho c6 nghiem la x = Bai tap phan Bai 2.1 Giai =+ 2cos^ 2) = (x^ - 2)e' + xe"^'"^ Bai Giai phuang phuang trinh trinh sau sau 3"^ (x -+ l)(x Bai Vai a, > 1, giai phuang trinh sau: (2' + x)(a" + 6') - 2(a + 6)' + x(a + 6) ^ Bai Giai phuang trinh sau 2^'^' + 2^"^^ = ;—^— log3(4x'-4x + 10) Bai Giai phuang trinh sau 2""''" + cos2x = 3"' + (2 + x' )'^''' ; X > U iJ-i^^ , Bai Giai bit phuang trinh x'' - 4x + > 2'"^'' ,r Bai Giai bat phuang trinh x(2" -1) + (2x -1)(9" - 3) + (3x - 2)(64' -16) > Huang dan giai bai tap phan Ta CO (x - l)(x + 2) = x^ - + X nen ta biln d6i phuang trinh nhu sau (x'-2)(e'-l) + x(e''-'-l) = -Neu x(x -2) = 0<» "x = thi ta thay phuang trinh thoa man 209 - N e u x(x^ - 2) 9t t h i chia ve cho xix^ - 2), ta dugc + X x^-2 = Ta thay rSng e'' -\a x luon cung dSu v a i nen vg trai luon duong va phuong trinh da cho truong hop la v6 nghiem Vay phuong trinh da cho c6 nghiem la x^O,x Xet phuong trinh 3' + 3"^ = cos^ Ta CO 3"" + 2cos^ x'^'-x^ > 2yfy~-3~' = ±42 x'^^-x = va dang thuc chi xay k h i x = H o n nua < va dang thuc xay k h i = k7r,k e Z T u day de thay chi c6 jc = la thoa man ca dang thiic tren Ta biSn d6i phuong trinh da cho n h u sau {2" + <:>x[a' + 6") = 2(a + by + xia + b) = 2' -^b'-a-b) (a + b^ ' z V I (a±b^ I 2 a'+b'\ J X n _ z V J r J X (2b 2a > ^ x\ ya±b) J Ta xet cac truong hop sau: N g u < ;c < t h i theo b i t d i n g thuc Bernoulli t h i ( 2a f 2a a+b a+b 2a - x + \ 2b 2b ^ 2b a+b a+b 2{a + b) < a+bJ a+b a+b - x + \n - x + = K h i do, \k phai duong, k h i a" +b' -a-b<Q nen ve trai am D o do, truong hop nay, phuong trinh v6 nghiem , NIU ;C<0 hoac x> \i cung ap dung b i t d i n g thuc B e r n o u l l i , ta c6 v8 phai t h i a ' + ' ' - a - < nen x^a" am T r o n g k h i do, neu x<Q +b''-a-b)>Q; 210 neu x>\i a' +b'-a-b>0 luon duomg , nen xia'+b''-a-b)>0, tuc la ve trai , Do do, phuong trinh truang hgrp ciing khong c6 nghiem Ngoai ra, ta th§y rang x = va x = nghiem dung phuong trinh da cho Vay phuong trinh da cho c6 nghiem la x = 0,x = \ Xet phuong trinh 2'"^' + 2'-'^ = — ^ log3(4jc'-4x + 10) Digu kien < 4x^ - 4x +10 7t 1, luon dung Ta CO 2''"' + 2'-^" > 2V2'-'.2'-'^ =8 va l o g ( x ' - x + 10) = log3((2jc-l)'+9)>log39 = nen 16 l o g ( x ' - x + 10) Do do, dang thuc phai xay hay r2x + l = - x 2A;-1 = Vay phuong trinh c6 nghiem la x = - Xet phuong trinh 2"'^'^ + cos2x = 3"' + (2 + f^"^ Taco cos^x<l va cos2x<l nen 2""'"^+ cos2x< + Ngoai ra, x^ > va |x| > nen 3"' +(2 + x^y^W > i + = Dang thuc phai xay nen x = Vay phuorng trinh da cho c6 nghiem nhdt la x = Xet b i t phuong trinh x^ - 4x + > 2'"^'' i i f L : f ^'^^•^ ^ " Taco x ' - x + = ( x ' ' - x + 3) + l = ( x - l ) ' ( x ^ + x + 3) + l > l Ngoaira 2^' > ° = l = > l - ^ ' < - = 0=>2'"''' < " = Do do, ta luon c6 x" - x + > 2'"^'' DSng thuc xay va chi x = Vay nghiem ciia bat phuong trinh da cho la r x^l Xet hk phuong trinh x(2' -1) + (2x-1)(9^ - ) + (3x-2)(64" -16) > Ta sg Chung minh r§ng x ( " - l ) > , ( x - l ) ( " - ) > , ( x - ) ( " - ) > That vay, xet bieu thuc x(2'' - l ) : Vai x < thi 2'-\<0 - Vdri jc>0 thi 2'^-l>0 nen x ( ' - l ) > - nen x ( ' - l ) > ^ Do do, ta luon CO x(2''-1) > vai moi jc e IR Xet bi6u thurc ( x - l ) ( ^ - ) : Vai x<^ - Vai x>^ - thi x - l > 0,9"-3 > nen ( x - l ) ( " - ) > thi 2JC-1 < 0,9"-3 < nen ( x - l ) ( " - ) > Do do, taiuonc6 (2x-1)(9"-3)>0 vai moi xeR Xet biSu thuc (3x-2)(64" -16): Vai x < thi 3x-2 < 0,64"-16 < nen (3x-2)(64"-16)> - Vai j c > | thi 3x-2>0,64"-16>0 nen (3x-2)(64"-16)>0 - Dodo, taluonco (3x-2)(64"-16)>0 vai moi jceM R6 rang dang thuc xay cac danh gia tren Ian luot tai x = 0,x =—,x = — nen khong the dong thai xay Vay bat phucmg trinh da cho luon dung vai moi x 5) Phuong phap dung ham so Nhan xet: Phirong trinh mu, logarit gidi quyet bang cdch dung ham so khd bien vd da dang, phong phu Cdc bdi todn dang co mot dgc diem chung la co so nghiem xdc dinh truac duac vd gid tri chinh xdc cua timg nghiem dugc nhdm dua tren phucmg trinh chic khong phdi thong qua mot phep bien doi dai so cu the ndo Ta phat b i l u hai dinh l i quan dung cac bai toan lien quan duai day - Dinh li Lagrange Cho ham s6 / ( x ) : [a, b]-^R lien tuc tren [a, b] va kha vi tren (a,b) K h i do, ton tai mot s6 thuc c e ia,b) cho / ' ( c ) = / ( ^ ) ~ / ( ' ^ ) , b-a - Dinh li Rolle Cho ham s6 f{x): [a,b] R lien tuc tren [a,b] va kha v i tren {a,b), d6ng thoi / ( a ) = / ( b ) K h i do, t6n tai mot s6 thuc c e {a,b) c h o / ' ( c ) = Mot so he qua thuang diing cua cac dinh li la: (1) N6u ham s6 / ( x ) : [ a , ] ^ E lien tuc tren [a,6] va kha v i tren {a,b) va phucmg trinh / ( x ) = c6 A: nghiem thuoc {a,b) thi f'{x) = Q c6 it nhit nghiemthuoc k-\ {a,b) (2) Ngu f(x): [a, 6] ^ R lien tuc tren [a, 6] va kha v i tren (a, 6), d6ng thai dao ham cap k cua ham / ( x ) khong doi dau tren f{x) = thi phuang trinh /(;c) = CO khong qua ^ nghiem phan biet thuoc (a, 6) Ta cung nhSc lai cac dang cua bdt dang thuc Bernoulli g i n liSn voi danh gia ham so mu la: (1) V a i 0<a^l thitaco a' >ia-Y)x + \, x<Ovx>l a" <(a-l)x 0<x<l + l, (2) V a i x>0 thi x" > a{x-1) +1 vai moi a > va dang thuc xay r a k h i sjC x= l Chung minh -2 (l)Xethams6/(x) = a ^ - ( a - l ) x - l Ta CO f'{x) =fl^In a - (fl -1),/"{x) = a" (In af > nen phuang trinh f{x) = CO khong qua hai nghiem phan biet Ta lai thiy /(O) = /(1) = nen jc = 0,x = la hai nghiem cua phuang trinh f(x)-O.Yebang bien thien, ta de dang thay dugc rang fix) > 0, X e (-oo; 0] u [1; +oo) va f{x) < 0, Vx e (0; 1) 213 Tir de dang suy bat dang thuc can chumg minh la diing (2) Ta xet ham s6 f(x) = x"-a(x-l)-\,x>0 Ta CO fix) = ax"-' - o = aix"-' -1), /'(x) = a{a - Ox""' > nen ham s6 da cho dat gia tri nho nhat tai nghiem ciia phucmg trinh / ' ( x ) = hay / ( x ) > / ( l ) = Do do, taduac x''>a(x-I) + l , V x > , a > l -' Tiep theo, ta se xet cac vi du su dung ham s6 de giai theo loai la true tiSp va gian tiep 5.1 Diing ham so ddnh gid true tiep Trong phdn nay, ta xet cdc bdi todn c6 dang / ( x ) = vd do, hdm f(x) CO dgo hdm cdp J, hoac dong bien Khi do, so nghiem cua phuomg trinh da cho dicac biet chinh xdc vd de hodn tat lai gidi, ta chi can nhdm cdc nghiem Vi dy Giai phuomg trinh sau 8'(3ac + l) = ' L&igiaL Taco 3x + l = — > nen x > - - Suyra 8^ ^ = * 8^ ^ 3x + l Xet ham s6 / ( x ) = 8"/ ' ( x ) = 8"hi8 + 12 (3x + l)- 3x + l VOI X ' > mdd sr«>-' ' in thi nen day la ham d6ng bien >0,X6 Do do, / ( x ) = CO khong qua nghiem Hon nOa, / ^ ^^ phucmg trinh da cho CO nghiem nhdt la ^ = ~Vi du Giai phucmg trinh (2 - x)(2 + 4") = LcigiaL Do + 4"" > voi moi x e R nen phucmg trinh da cho tuomg duomg voi 214 — ' " ^ ^ + 4' (2 + 4')' /'(0 = 0«4"+4'(4-6.1n4) + = Day la phuong trinh bac hai theo bi^n 4' nen no c6 khong qua hai nghiem Do do, phuong trinh f'{t) = c6 khong qua hai nghiem (m6i gia tri duong cua 4' cho ta dung mot gia tri cua t) Tu do, ta thay phuong trinh / ( / ) = c6 khong qua ba nghiem Mat khac ta cung c6 /(O) = /(1) = / ( ^ ) = nen cac gia tri cung nghiem dung phuong trinh ban dSu Vay phuong trinh da cho c6 ba nghiem la x = 0,x = \,x = ]- , ^ ,^ Vi du Giai cac phuong trinh sau a) r^^'cosX + 2cos'x = 2"'^'^"' + 4cos'jc ,j - b) 3^=l + x + log3(l + 2x) f LMgiaL a) Ta c6 2""'^ cos X + cos' X = 2*^°''"-' + 4cos' x « 2'=°''^-' (2 cos x -1) = cos' x(2 cos x -1) o (2 cos X -1)(2"'''^ - cos' x) = o cos x = ^ v 2'=°''^ = cos 2x +1 -Voi cosx = - , t a c x = + —+ A:2;r,^eZ T -Vai 2""'"" = cos 2x +1, ta thSy day chinh la phuong trinh c6 dang Bernoulli \ nen no c6 nghiem la cos2x = 0vcos2x = <=>x = — + A:—vx = kn,keZ Vay phuong trinh da cho c6 ba ho nghiem la X = ±—+ A;2;r,x = — + k — ,x = k7i,k e Z 215 b) Dieu kien xac dinh ^ > ~ " Phuong trinh da cho tuong duong vai 3' + x = 3'°^'*^''^" + logj(2x +1) D I tli4y ham s6 / ( / ) = 3' + / e M d6ng b i l n tren R va phuomg trinh tren chinhla / ( x ) =/(log3(2x + l)) nentaducrc x = log3(2x + l ) o ' ' = x + l Phucmg trinh c6 dang b i t ding thuc Bernoulli nen no c6 nghiem la x = 0,x = \a day ciing chinh la hai nghiem ciia phuomg trinh da cho V i du Giai phuong trinh 5" + 4^ = 3" + 2" +1 Ox' - 6x L&igidL Xet ham s6 f(x) = 5" +4' - (3^ + 2^ + lOx' - 6x), xeR Ta tinh dugc / " ( x ) - 5" (In 5)' + 4" (In 4)' - 3" (In 3)' - 2" (in 2f Ta thay x < khong phai la nghiem ciia phuong trinh da cho v i nSu ngugc lai thi 5^ + 4' < 3^ + 2MOx' - 6x > => 5' + 4^ < 3' + 2' +1 Ox' - x , mau thuln Do x > = > / ' ' ( x ) > Suy phuong trinh / ( x ) = c6 khong qua ba nghiem phan biet Hon nua, ta ciing c6 /(O) = /(1) = / ( ) = nen phuong trinh da cho c6 dung ba nghiem la x = 0, x = 1, x = 5.2 Dung ham so de ddnh gid gidn tiep Cdc bdi xet cdc vi du ducri day tuang doi kho horn va tinh dcm dieu cua cdc ham chua duac nhln nhdn tic ddu, thdm chi, ta chua tim duac bieu thuc tuang minh cua ham cdn xet ma phdi thong qua mot so buac bien doi V i du Giai phuong trinh sau 2'' + 3'' = 2" + 3"^' + x + LdigidL Phuong trinh da cho tuong duong voi ( ' ' - ^ " ' ) + (3''-3^"') = x + l - ^ Ta xet cac truong hgrp sau - NSu 2^ > x + thi 2'' - 2"*' > 0; ' ' - 3"^' > 0; x +1 - 2" < nen phuong trinh khong thoa man 216 - Nlu 2^ < X +1 thi 2'' - 2^"' < 0; 3'' - 3^"' < 0; x +1 - 2^ > nen phuong trinh khong thoa man - N8u 2"^ =x + l thi phuomg trinh da cho thoa man va do, nghiem ciia no cung la nghiem cua 2"^ = x +1 Day la phuomg trinh dang Bernoulli nen c6 nghiem la x = 0,x = l Vay phuong trinh da cho c6 hai nghiem la x = 0,x = Vi du Giai phuong trinh 2014'' (Vx^ +1 - x] = Lcri gidL Chu y rSng Vx^ +1 - x | Vx^ +1 + x j = nen ta c6 2014^ =-p=L— = y[7+l+x Vx'+l-x 2014-"=^=i va = Vx^+l-x »^ Vx'+l + x Tu day suy 2014^-2014-^ = 2x Xet ham s6 /(x) = 2014" - 2014"" - 2x, x € E , ta CO fix) - (2014" + 2014"') In 2014 - > In 2014 - > 0, theo hk dSng thuc Cauchy Do do, /(x) la ham dong bien nen /(x) = c6 khong qua nghiem Ta lai c6 /(O) = nen phuomg trinh /(x) = c6 nghiem nhat la x = Thu lai vai phuong trinh ban dku, ta thiy thoa man Vay phuomg trinh da cho c6 nghiem la x = 5.3 Dung dinh liLagrange v&i cdc phirang trinh co dang f{a) = f (b) Trong dang nay, ta cung se xet mot ham so de giai quyet nhieng khong phdi la bien x thong thuang ma bien duac chon dua vao cdc he so hogc ca so cho truac, bien x duac xet dual dang tham so Day la mot dang phuong trinh thu vi va each dung dinh li Lagrange de giai quyet la hieu qua nhdt 217 Vi du Giai cac phuomg trinh sau „ " •^1 - a) 3" + ^ = - \ b) 9^(3^+2^) = 2^(8'+ 7^)+ 5^(5^-2^) • r L&i giaL a) Phuomg trinh da cho tuomg duomg vcri 3"^ - S'' = S'' - 7"^ Xet ham s6 / ( / ) = - (/ + 2)'", la nghiem cua phuomg trinh tren, suyra/(3) = / ( ) Ta thay ham so lien tuc tren R nen no cung lien tuc tren [3;5], theo dinh l i Lagrange, ta c6 /(3)-/(5) 3c €(3; ) : / ' ( c ) = • = hay xo (c"""' - (c +1)"""') = 3-5 Tir ding thuc nay, ta thay = v XQ = 1, tuc la nSu x^ la nghiem cua phuong trinh da cho thi Xg = v x,, = Thu lai, ta thiy ca hai nghiem d^u thoa man ' ' ' '^ Vay phuomg trinh da cho c6 hai nghiem la x = 0, x = b) Phuomg trinh da cho tuomg duomg voi 9" 0" + 2^ ) = 2^ (8^ + 7^ ) + 5" (5* - 2" ) 10^ - ' - 25' = 14' -18" - 27' « 10'+12'-16'-25'=12'+14'-18'-27' '* (*) Gia su Xp la nghiem ciia phuomg trinh (*) Ta xet ham so sau f(t) = /'° + (/ + 2)'" -(t + 6)'° - (/ +15)'°, / > D l t h d y (*)<::>/(10) = / ( ) Ham s6 / ( / ) lien tuc tren doan [10,12] nen theo dinh l i Lagrange thi t6n tai s6 Q thuc c e (10,12) cho / ' ( c ) = 1u 10-12 Do do, neu x^ la nghiem cua phuomg trinh (*) thi no phai thoa man c'»-' + (c + 2)'"-' - (c + 6)'"-' - (c +15)'°"' = <:> Xo = V c'"-' + (c + 2)'°"' = (c + 6)'"""' + (c + 15)^"' 218 Ding thuc thu hai cho ta = vi n l u fXu - f , X ( , < l = > X o - l < ^ c ^ " ' + ( c + 2)^°"' >(c + 6)^~'+(c + 15)^"',tuomgtu >1 cung mau thuan Thu lai true tiep, ta thay hai gia tri x = 0,x = thoa man phuomg trinh da cho Vay phuong trinh da cho c6 hai nghiem la jc = 0,x = Vi du Giai cac phuorng trinh sau r a) x'°^^'-10"'^^=2x ,•- [ -J b) 9"'"-4'"^^=2cosjc L&igidi a) *)Dieukienxacdinh: 0<x^\ r Phuong trinh da cho tuong duong voi ; v -I'l ;i : Gia su phuong trinh c6 nghiem la x = x^ Taxethams6 s a u / ( = (^ + 4)'°«^'°-^'°«''°,r>0 - Phuong trinh da cho chinh la / ( ) = / ( ) Ham s6 lien tuc tren [2,4], kha ' vi tren (2,4) nen theo dinh l i Lagrange thi ton tai so thuc c e [2,4] thoa man / ' ( c ) = -^^^^ -^^^^ = « (log, x,)Uc + 4)'°^^-' -c'"^^^-' = 6—2 De thay dang thuc tren cho ta = v x^, = 6, day chinh la dieu kien can de ' X = JCQ la nghiem ciia phuong trinh da cho Thu hai gia tri true tiep vao phuong trinh da cho, ta thay thoa man Vay phuong trinh da cho c6 hai nghiem la x = 1, x = b) Phuong trinh da cho CO the viet lai la 32COS _ 22COS = 3.2 COS X - • COS X o ' " " - • cos x = 2'^°^^ - • cos x Dat f = 2cosx thi 3' -3r - ' - I t (*) Gia su phuong trinh c6 nghiem la / = Ta xet ham so / ( a ) = ấ" -at^, phuong trinh tren chinh la / ( ) = / ( ) Ham so lien tuc tren [2,3] va kha ' vi tren (2,3) nen theo dinh l i Lagrange, ton tai c e (2,3) thoa man 219 /'(c) = / ( ^ - p i = « /„c'»-' - /o = « ^0 (c'"-' -1) = D I thdy rang ding thiic tuomg duong voi = v = 1, day chinh la digu kien can de t^t^ la nghiem ciia phuong trinh (*) Thu true tiSp hai gia tri vao, ta thdy (*) thoa Do do, (*) CO hai nghiem \a t = 0,t = \a tuong ung, ta c6 cosx = v cosx = ^ Giai cac phuomg trinh lucmg giac nay, ta thu dugc cac nghiem cua phuomg trinh da cho la x = — + k7r,x = ±— + k27r,keZ Nhgn xet: Ta thdy rSng phuomg phap dung dinh li Lagrange nhu thg rit hieu qua doi voi cac bai toan dang Tuy nhien, a buac tim dieu kien can cua nghiem phuong trinh da cho, neu phuong trinh c6 dao ham bing voi an c neu tren khong c6 nghiem khong phu thuoc c thi each giai khong thk su dung duoc va bai toan can phai ap dung phuomg phap khac dk giai quylt.' Bai tap phan Bai Giai phuong trinh sau 4"^ = x + Bai Giai phuomg trinh sau 9" + 3" = (2JC +1) • 2"^' Bai Giai phuomg trinh sau 2"" +5'' = 25x + 27x + -5" Bai Giai phuomg trinh sau 12^ +13^ +14^ = 2^ + 3^ + 4^ + 870x^ - 2400x^ +1560x Bai Giai phuomg trinh sau yjs" -2x + yJ2x + \ +4x + \ 5^'' Bai Giai phuomg trinh sau + = (1 + 2VJc)'°^'^ Bai Tim tat ca cac so duong o ma bat phuomg trinh sau nghiem dung voi moi jc e M a^>x + l Huang dan giai bai tap phan Xet ham s / ( ; c ) = 4" - X - voi x € R Ta CO fXx) = 4'\n4-\a / ' ( x ) - 4"(ln4)'> Do do, phuomg trinh f{x) = c6 khong qua nghiem Ta lai c6 /(0) = / 2) = nen phuong trinh da cho c6 nghiem la x-0,x = — Phuong trinh da cho tuong duong vai 2) Xet ham s6 / ( x ) = = 4x + « — v2 <9> /V) = .2, + r9> / -> \ + + — - x - = - 4x - 2, X e R thi tucmg tir tren, ta c6 + '3^ '3^ .2, .2, > nen phuong trinh c6 khong qua nghiem Ta lai c6 /(O) = /(1) = nen phuong trinh da cho c6 nghiem la x = 0, x = Xet phuong trinh 4" + 3' = 4x + + log^ (Sx + - 3^) Dieu kien xac dinh 5x + > 3"" Dat / = log4 (5x + - S'') <=> 4' + 3"^ = 5x + 2, ma thay t vao phuong trinh da cho thi ta CO 4^+3^ =4x + + / So sanh hai dang thuc nay, ta dugc 4' - 4"" = x - / <» 4"" + x = 4' + ? Cung tir tinh dong bien cua ham so f(y) = 4^ + y, ta c6 x = t hay 4"" + 3"" = 5x + Phuong trinh tuong duong voi (4'' - 3x -1) + (S'' - 2x -1) = Theo hk dSng thuc Bernoulli thi hai b i l u thuc ngoac cua v6 trai luon cung dau voi nen dang thuc xay x = 0,x = 1, thoa man dieu kien xac dinh Vay phuong trinh da cho c6 nghiem la x = 0, x = 221 Xet ham s6 / ( x ) = 12" +13" +14" - (2" + 3" + 4" + 870x' - 2400x' +1560x) Ta CO (x) = 12\(ln 12)' +13^(ln 13)' +14\(ln 14)' - 2^(ln 2)' - 3".(In 3)' - 4\(ln 4)' Ta thay rang phuong trinh da cho, x phai la s6 khong am vi neu ngugc lai thi theo bat dang thurc Bernoulli thi 12" > l l x + l;13" >12x + l;14" >13x + l /^ ^ ^ ' " ^ Suyra 12'+13^+14" >36x + Tuy nhien, ta ciing c6 2^ + 3" + ' + 870x' - 2400x' +1560x < + 36x + x(870x' - 2400x +1524) <3 + 36x Do 12' +13" +14" > 36x + > 2' + 3' + ' + 870x' - 2400x' +1560x, khong thoa man Tir ta dugc x > va ham so dang xet a tren thi / ' " ' ( x ) > f t r Dao ham cap khong doi dau nen phuong trinh / ( x ) = c6 khong qua nghiem Mat khac, ta cung c6 /(O) = / ( I ) = / ( ) = / ( ) = nen phucmg trinh / ( x ) = CO dung nghiem la X = 0,x = l,x = 2,x = Xet phuong trinh 4^-1% + V2x + + 4x • 5" + 4x +1 = Dieukien 5' > x , x > - | Dat o = V " - x , = V2x + l , a , A > T a c a' = 5" - 2x,6' = 2x + a' - 6' = 5" - 4x - l , a ' + 6' = 5" + Dodo (a' + - ^ ) = (5"^ +1)(5" - x - ) = 5'" - 4x • 5" - x - Phuong trinh da cho c6 the viet lai la a-b = {a'+b^ ){a^ -b^)<^{aa-b = 0<^a = b ihi 5'-2x b){\ (a' + b^ ){a + = 2x + l^5'=4x b))^0 + l 222 Day la dang phuomg trinh c6 dang Bernoulli nen no c6 hai nghiem la jc = 0, X = 1, thoa man dieu kien de bai „^ , - N6u (a' +b^)(a + b) = l^ (5^ + 1)(V5^-2JC + V2x + 1) = Ta thay rSng V s ' - x + V2x + > ^ ( ' - x ) + (2x + l) = Vs^+1 > va 5^+l>l nen phuomg trinh tren v6 nghiem Vay phuomg trinh da cho c6 hai nghiem la x = 0, x = Xet phuomg trinh sau + Vx = (1 + lyfxj"^'^ Di6u kien x > Ta c6 1+ = (1 + 2^p^^ «1 + = 2'°^'('*'^) « log, (l + V ^ ) = log3 (l + V ^ ) Dat log, (l + V ^ ) = log, (l + V ^ ) = t thi (;: - ^/^ + l = 2',l + 2^/^ = ' ^ - ' = ' + l c ^ ' - l = 3'-2' Xet ham s6 /(M) (M +1)'" - u'", la nghiem cua phuomg trinh tren, suyra/(l) = /(2) Ta thdy ham s6 lien tuc tren R nen no cung lien tuc tren [1;2], theo dinh li Lagrange, ta c6 3ce(l;2):/'(c)=^^^^^^5|^ = hay (c'-'-(c + 1)'-') = ^ nsi Tir dang thuc nay, ta thay = v = 1, tuc la neu la nghiem ciia phuomg trinh da cho thi = v = - Vori/ = 0, taco log2(l + V x ) = 0<::>l + >/x =l<;=>x = - Vai / = l,tac6 log,(l + >/x) = l«1 + V x = < = > x - Thu lai, ta thay ca hai nghiem deu thoa Vay phuomg trinh da cho c6 hai nghiem la x = 0, x = Xet bdt phuomg trinh a"" > x + l (chu y la bk phuomg trinh c6 dang Bemolli a" >{a- l)x +1 nhung doi hoi dung vai moi x nen diSu kien cua a phai manh hom ciia dieu kien ciia bat dang thuc Bemolli, tuc la a > 2) 223 Ta se tim dieu kien can va du cua a Ta thay neu jc < -1 thi bat dang thuc dung nen chi cin xet x > - Bat ding thuc da cho tuong duong vai x\na>ln(x + l) (*) Ta CO truong hop: - Neu X > thi Ina> ^ ^ ^ ^ ^ Ta se chung minh rSng ^"(•^"*"<\l jc> X Thatvay, xet / ( x ) = ln(x + l ) - x , x > thi f\x) = — x+\ x 1=^ < nen fix) nghich bien Suy / ( x ) < /(O) = Do do, dh bdt ding thuc (*) dung vai moi jc> thi ta can c6 In a > <=> a > e -N^u - l < x < thi l n a < ^ ^ ^ ^ ^ ^ ^ -ln(x + l) Xet ham s6 g{x) = Mf±ll thi g'(x) = ^ ' X x' ' Xet tigp hix)^——ln(x + l) thi h'{x) = — ^ — x + \ + \y x + \ + \y x € ( - l ; ) nen h(x) d6ngbi6n —= , >0 Suy h{x) > h{0) = 0, dan den g'(x) > nen g{x) nghich bign va , , ln(x + l) , g(x)>lim^ ^ = Do do, dk hk ding thuc (*) dung vai moi - I < x < thi ta can c6 In a < <=> a < e Tu hai truong hap tren, ta thiy chi c6 gia tri a = e la thoa man Vay vai moi x e M, ta luon c6 bat dang thuc e"" > x +1 6) V i du tong hgrp Trong phdn nay, ta se xem xet mot so bai yeu cdu bien doi cdc dang thuc hogc cdc dang phuomg trinh mil va logarit khong mdu muc 224 Vi du Cho X, y, z la cac so thuc thoa man 2" =3^ = 6'' Tinh M = xy + yz + zx L&igidL Ta thay neu mot ba so x,y, z bang thi hai s6 lai cung bSng va M = Xettruomghgp xyz^O, dat 2' =3^ =6'' =k>0 i _1 Khido = it^3 = )t^6 = / t ' ^ m a - - nen i i _ i i i 1 k"!^ =k - <=> A:'' ^ = k -« — + — = — <=> xy + >'z + zx = X y z Vay moi tmong hop, ta deu c6 M = Vi du Chung minh phuomg trinh (x +1)' = x''"^' c6 dung mot nghiem duong L&igidL Phuong trinh da cho tuomg duomg vai X _ x+1 Inx ln(x + l) Xet ham s6 f(t) = ^ c6 f'iO = nen ham so c6 dilm cue tri la / = e R5 rang x +1 > x nen de c6 dugc / ( x +1) = /(x) thi ta can c6 X6(0,e), x + e (e,+oo) \nt Do thi ciia ham so /(O = — dat cue dai tai x = e nen duong thang y = m,e>m>0 nam ngang cat thi tai dung hai diem va tinh lien tuc va dom dieu tren cac khoang nen ton tai dung mot gia tri m ma khoang each giila hai giao diem noi tren la Suy c6 dung mot gia tri x cho x''^' = (x +1)'' Bang each nay, ta ciing chung minh dugc vai mgi a > thi phuong trinh x"" = (x + ay c6 dung mot nghiem thuc xe{0,e] 225 Vi du Goi T la tap hop cac cap so thirc duofng a,b phan biet thoa man dSng thuc ct =}f Chung minh rang ton tai nhdt mot s6 thuc khong phu thuoc a,h cho gia tri nho nhat cua ham so x-a + X - 6|+|x - 2|+|x - 3| dat tai /^^ {x^) vai mpi cap gia tri (fl,6)€r Lcfi gidL Khong mk tinh t6ng quat, ta gia su a<b, theo nhan xet tren thi 0<a<e<b Theo bat dang thuc ve gia tri tuyet doi, ta c6 x-a + x-b + x-2 + x-3 > {x-a) + {b-x)\ \(x-2) + {3-x) b-a+l DSng thuc xay Ichi va chi {x-2)(3-x)>0,ix-a)ib-x)>0^2<x<3,a<x<b Tuy nhien, ham so xet moi cap (a,b)eT va b-a-^0 thi x - > e Do do, gia tri XQ de bai neu chinh la e Vi du Giai phuong trinh nghiem nguyen sau \0 + \V +6" =^yj3j L&igidL a) Do ve trai nguyen nen ve phai ciing phai nguyen, suy ^.^ y\>\<i:> y > Mat khac, ta thay (Vs)" =11 + 10'+ 6' >10 =>>'!>2 =>>'>3 =>>-!>6 =>(V3)'">27 - N6u >' = thi 11 +10' + 6' = 27 «10' + 6' = 16 ve trai tang theo biln x nen phuong trinh c6 nghiem la jc = - Neu y>3 thi (73) chiahet cho va de thay x > Ta c6 « ll + ' + ' = l l + (9 + l ) ' + ' = l l + ^ C ; ' - ' + ' =12 + | ; c ; ' - ' + /=0 /=0 D I thiy rang ^C^9''"' +6"^ chia hk cho va ve phai cung chia hk cho <=0 nhimg 12 khong chia het cho nen tniong hgfp phuong trinh khong CO nghiem nguyen nao thoa man Vay phuong trinh da cho chi c6 nghiem la x = , = Vi du Hay xac dinh tat ca cac cap so duong a,b cho voi moi so nguyen ducmg n va vai moi nghiem thuc x„ ciia phuong trinh W x = log2(2«^x + l ) , ta luon CO a""+6"" >2 + 3x„ L&igidL Dieu kien xac dinh cua phuong trinh la x > — ^ In Phuong trinh da cho tuomg duong vai 2""'' = In^x +1 <=> 2*"'"^^ = 4n^x + Dat / = 2n^x +1, ta c6 2' = / +1 Day la phuong trinh Bernoulli nen c6 hai nghiem la ^ = 0,/ = l Suy phuong trinh da cho c6 hai nghiem la x„ = 0,x„ = - 4n2 • Xet bat ding thuc a"" +b"'- >2 + 3x„ Dieu kien 0<a,b^l • -i? - Voi x„ = thi a''" + 6''" = = + 3x„, bit ding thuc dugc thoa man voi mpi a,b -Voi x„ = — ^ , b4t ding thuc da cho tro 4n' a +b ^ L L ^ >2—^<:> a'"' +b'"' Cho « -> + 0 , ta biet rang lim a-'+b J t _ 4«' = yfab nen ve trai c6 gidi han la 4ab , ve phai c6 gidi han la eyje 227 Bit ding thurc phai dung vai moi n nen yfab <e4e <^ ab<e^ Ngugrc lai, vai ab < thi (do ta CO bdt ding thuc a +b ^ \- ; > + 2- V 4« y > 2" > + x, Vx e M ) Suy a"" +b''" >2 + 3x„ cung dung Vay digu kien c i n tim \a < a,b ^ \ ab < Vi du Chung minh r i n g nSu A,B,C la cac goc cua mot tam giac thi phuomg trinh sau luon c6 nghiem phan biet 3^" ^'^ = sin— + sin— + sin — 2 L&i giai a) V a i A, B, C la ba goc cua tam giac thi < sin ^ + sin Thatvay,dat X = ^ ^ , F = ^ ^ , Z = ^ ^ • 2 X + Y + Z^;r thi 0<X,Y,Z<n: + sin ^ ^ ^ • va nen cac goc X, Y, Z cung la ba goc cua mot tam giac Ta CO / Z ^ A B C sin — + sin — + sin — = cos Z + cos F + cos Z = - sm 2 2 Do do, ta xet phuang trinh an jc c6 dang 3 +- < - 2 r2j x^-2x Khao sat ham s6 tren miSn tuomg ung, ta c6 diSu phai chung minh Vi du Tim tdt ca cac gia tri a dg bdt phuang trinh sau nghiem dung vai moi x L&i giai Dat ^ = S'' > Bat phuang trinh da cho tra fl/^+9(a-l)/ + a-l>0<::>a(/^+9/ + l ) > / + l < » a > 9/ + /'+9/ + 228 Bit phucmg trinh da cho nghiem dung vai moi x va chi 9^ + 1 a > max Xet ham s6 / ( / ) = ^^^^,t t'+9t + \ Ta CO /'(O = -9r-2t < 0, > > nen day la ham nghich biSn Suyra/(0</(0) = l Dodo 9^ + < 1, > nen cac gia tri a can tim la a > t'+9t + \ Vi du Giai cac bit phuomg trinh sau a) >x^ vai X la so duong khac b) c o s ' - - + 1> ^ cos ^ cos — L&igidL a) Ta c6 biSn d6i x ^^"'''^^ >x\Ta xet cac truomg hop: -14 -Neu jc> thi taphai c6 -{3x + 7)>7<^x<, khongthoaman 14 -Neu 0<x<\i ta phai c6 -(3X + 7)<7<^3X + > - < : : > X > - Y , dung Vay nghiem cua bit phuomg trinh da cho la x e (0;1) b) Ta CO bien doi: COS — 4cos + ^ COS — 2cos^ rCOS 3;r^ — + COS — I j I 7; , JC > COS I — ) , a i « 37r 27r n De thay rang < cos— < cos— < cosy < •^•x 229 Do do, ta CO tnrong hop: -Neu jc>0 thi nY cos— > I COS — nen bat phuong trinh nghiem dung, ) ( 3;r^ f 2;r^ nen bdt phuong trinh cung nghiem dung - NSu jc < thi > cos— cos — V ; I J Do do, nghiem cua bat phuong trinh da cho la x e R V i du a) Giai bdt phuong trinh e"^ + cos JC > + JC - y b) Giai bat phuong trinh 2^"'+6JC-11 jc-2 >4 (Du hi DH khdi B 2003) (Die bi DH khdi B 2004) c) Chung minh r i n g phuong trinh sau c6 dung nghiem phan biet e'' -sinjc + — = (Du bi DH khdi B 2004) L&igidL a) Ta se chung minh rang voi cosx > —— voi moi JC That vay, neu thay x boi -JC thi b i t dSng thuc khong d6i nen ta c6 th6 gia su JC > JC^ h Xethams6 / ( j c ) = cosjc - + — , x e M ta c6 / ' ( x ) = - sin x + x va ' /''(x) = l - c o s x > Do / ' ( x ) d6ng bien va ta c6 / ' ( x ) > / ' ( O ) = 0, dan den / ( x ) d6ng biSn va / ( x ) > / ( ) = Chii y rang ta cung c6 e"" > x + voi moi x nen cong hai bat dang thuc lai, ta dugc e'' + cos X > + X - ^ Bat phuong trinh da cho c6 nghiem voi moi x b) Dieu kien x^2 Bat phuong trinh da cho tuong duong - — — - > x-2 Chu y rang phuong trinh 2"'^ + 2x = c6 v§ trai tang va x = thoa man nen no CO nghiem nhat la x = 230 Dithay ^ ' ' + x < < » x < l va ' " ' + x > < » x > l Xet truong hop: -Neu - Neu 2"-'+2x-3>0 x-2>0 < fxf x > l <=> S o x>2 X > fx<l 2^-'+2x-3<0 <» ^ x-2<0 » X < [x<2 i Vay bat phuang trinh da cho c6 nghiem la x e ( - < » ; 1) u (2; +QO) c)Xethams6 f(x) = e''-smx + ——3 thi / ' ( x ) = - cos x + x va / ' ( x ) = e'' +1 + sin X > nen phuang trinh / ( x ) = c6 Ichong qua nghiem Chii y rang ham s6 / ( x ) lien tuc tren M va lim / ( x ) = +00,/(O) = -2 < 0, lim / ( x ) = +oo nen phuang trinh da cho c6 dung nghiem " Bai tap tong hgp Chuang Bai Giai cac phuang trinli sau a) 5^+4^+3^+2^= — + — + — - x ' + x ' - x + 16 2" 3" 6' ' b) 6^+1 = 8^-27^-' Bai a) Cho X la nghiem cua phuang trinh (3 + 272/ =(N/2-1)"+3 • ^ i Chung minh rang x ciing la nghiem cua phuang trinh (V2 +1)"" = 2cos-^ • ^ ' b) Biet rang so thuc a thoa man phuang trinh 4"^ = cos ox c6 dung 4"^ 2014 nghiem thvrc Hoi phuang trinh 4"^+-^ = (cos ox+ 2) c6 bao nhieu nghiem thuc? Bai Giai phuang trinh 2^'"^' + l o g ^ x = V 231 Bai Giai cac pliuang trinh sau a) 5^ + 5^' = 4^ + 6"' (Di thi chon doi tuyen DHSP 2010) b) 3" (4" + 6" + 9") = 25" + • 16\ chi Todn hoc tuSi tre 2008) Bai Giai cac phuong trinh sau: a) fix) = voi f{x) = x + e'' -e\ b) (x + 6) • 5'"i""'i - X = (x +1) 5" - + 5"^' +1 (De thi chon doi tuyen DHSP 2011) Bai a) Giai phuong trinh sau = o^^-'^ -a^cos^- +^2sin^x + _ ( ^ + ^)2cos^ _^2cos^ ^ ^ ^ ^ ^ ^ tham so ion hon b) Tim a de phuong trinh sau c6 nghiem nhat 9"" + = o • 3"" cos [nx) Bai Giai phuong trinh sau a) j c " ' - ' + ( x ' - l ) " + l - x - x ' = b) 3''"' + 3"'' = 2'^' + 2-^" + Bai Giai cac bat phuong trinh 12^+16^-18^-32" ^ - V 27"+36"+48"+64" " yfl^ ' b) 3''"'-"-36.3"-'+3>0 ^ 2x + l ^ ^ ^ 2"-l Bai Giai cac bat phuong trinh sau a) [x'°'''+x^'"' > 2012'"+2012" ^4022 ^ ^ ^2011'"+2011" , 6-3" 10 b) > vol X X 2x-\ > Bai 10 Giai cac bat phuong trinh sau: a) 3"'-'+(x'-4)3'"-'>l b) Xac dinh cac s6 thuc a > cho a'°''" > cos' x, Vx e 232 Huang dan bai tap tong hap chirang Bail a) Xet ham s6 1 n / ( x ) = (5^ + ^ + ^ + ^ ) - — + — + — + x ^ - x ^ + x - , x e M 2^ 3^ 6^ Ta thay rang n&u x<0 thi / ( x ) < nen phuong trinh khong c6 nghiem Ta chi xet x<0 x>0 Taco fix) = (5Mn + 4Mn + 3Mn + 2Mn 2) - ln2 ln3 •+ ln6 + + 12x'-4x + l > 6' J Vx>0 Suy ham s6 dong bign va phuong trinh f(x) = c6 khong qua mot nghiem Ta thiy / ( I ) = + + + - 1 —+ - + — + - + 1-16 = Vay phuong trinh da cho c6 nghiem nhSt la x = b) Phuomg trinh 6"^ +1 = S'' - 27'"' c6 thg vigt lai la (3-' f + (_2- )3 +1^ = • 3"-' • {-2') • a+b+c^O Mot ket qua quen thuoc la a +b +c = 2>abc <=> a =b=c vol moi a,b,csR Do do, tir phuong trinh tren, ta dugrc 3"^"'-2"^+1 = < » ' + = 3-2" hoac ' - ' = - ^ = (khong t6ntai x ) Ta se chung minh phuong trinh 3"^ + = • 2"^ chi c6 khong qua hai nghiem That vay, xet ham s6 / ( / ) = ' + - - ' , r e M Ta c6 / ' ( O = In • 3' - • In • 2', f'{t) = « In • 3' - • In • 2' = 31n2 v2y 233 Dao ham ciia ham s6 c6 nghiem nh4t nen ham s6 / ( / ) d6i dku khong qua hai \kn, tiic la phuomg trinh f[t) = c6 khong qua hai nghiem Ta lai thay rang / ( I ) = / ( ) = nen t = \,t = nghiem dung phuomg trinh /(0 = o V r ' Vay phuomg trinh da cho c6 dung hai nghiem la x = \,x = Bai2 a)Tathay + 2V2 = (V2 +1)' = (V2-l)^ nen neu phuomg trinh da cho, dat/ = (V2+1)">0 thi ^ ' = ^ + « / ' - r - l = ( * ) Ta se tim nghiem thuc duomg cua phuomg trinh Truac het, ta xet cac nghiem thuoc [-2; ] , dat r = cos or, or e [0,;r], ta c6 cos^ a - cos or - = >» cos 3a = — <:>3a = ±— + k27r<^a^±— + k—,keZ De thay rSng doan [0,;T], C6 gia tri a thoa la -^^-^r^^ tiic la phuomg trinh (*) c6 ba nghiem la c o s ^ , c o s - ^ , c o s - ^ Phuomg trinh (*) la phuomg trinh bac ba nen c6 khong qua ba nghiem, suy day la tat ca cac nghiem cua (*) ' *- ^t"^ Trong cac gia tri nay, chi c6 2cos^ > nen suy t = (•j2+\y = c o s ^ b) Phuomg trinh thu hai c6 the viet lai la 4' +-—2 = 2(cosax + l ) < » 2"- ^ - — = 2cos— 2' = 4cos —<=> 2' ]_ A ^ (1) ^ ' ox ax _ _ = _ c o s y (2) 234 Gia su 2014 nghiem phan biet ciia phuomg trinh x^,x^,x^, ,x2o^^ ' ' - ^ = 2cosax (*) Ta thdy ring phucmg trinh (*) khong c6 hai nghiem nao d6i vi ndu ngugc lai, t6n tai jc,,-jc, dSu la nghiem cua (*) thi ta c6 he sau • " -iSo - - i = 2cosax, =>4"—i- = 4-'—!-o4''=4-''ox=0,voll - ^ - - = 2cos(-ax,) - Xet phuomg trinh (1), dat ;c = 2/, ta c6 4' - - ^ = 2cosậ Theo gia thilt thi phuomg trinh c6 2014 nghiem phan biet la trinh (1) tuomg ung c6 2014 nghiem phan biet la x^,X2,x^, ,x2o^^ nen phuomg 2x„2x2,2x^, ,2x2^^, - Xet phuomg trinh (1), dat x = -2t, ta cung c6 phuomg trinh 4' - ^ = cos ' Theo gia thiet thi phuomg trinh c6 2014 nghiem phan biet la X|,jr2,x3, ,X2(,i4 nen phuomg trinh (2) tuomg ung c6 2011 nghiem phan biet la —2x^,-2x2,~2x^, ,—2x2()^^^ Do cac gia tri x^,x2,x^, ,X2o^^ phan biet va khong c6 hai s6 nao ddi nhu chung minh a tren nen cac gia tri -2x,,-2^2,-2x3, ,-2x20,,,2x,,2^2,2x3, ,2JI:2OI4 Vay phuomg trinh 4"^ + cung phan biet = 2(cos ox + 2) c6 dung 4028 nghiem phan biet, Bai Xet phuomg trinh 2"'^"' + log^ x = 2'*' Dieu kien x > Ta CO r'''' « <:A.+ log^ X = 2"' T'^'" « 2"'^"' + log2 x' = = Y^' + log2 X « Y'^'^ Y'' + log2 (x' + X - ) = Y^' + \og2 (X +1) * ~i~ 235 Xet ham s6 / ( / ) = 2' + log^ t,t>0 thi /'(O = 2' In2 + — > /In2 Do / ( la ham so dong bien va phuong trinh da cho chinh la /(x^ + ) = / ( j c +1) o + jc^ - jc + « (>: +1)^ (jc -1) = Do X > vay nen ta chi nhan nghiem x = Bai4 ' a) Phuong trinh da cho tuong duong voi 5"" - 6"' = 4" - 5"' Gia su X = XQ la nghiem cua phuong trinh tren Ta xet ham so sau f{t) = f'' -{t + \f' voi / > ! Taco f'{t)^x/''-'-xlf^-'=x/^-' Phuong trinh da cho chinh la / ( ) = / ( ) nen theo dinh ly Lagrange, ta thay rang t6n tai s6 thuc t e (4; 5) cho /'(O = -^^"^^"-^^^^ = 4-5 Xo=0 Ta thdy r i n g / ' ( / ) = « Xet tiep dang thuc x^f^''^ = 1, de thay x^ > va x^ = thoa man NSu Xo > thi xl - X Q > hay f'"" > /" = 1, suy xy''"' > NIU 0< XO < thi Xo - X o < hay f''-'' < /" = 1, suy XQ/""""" < Do do, voi moi t > thi dang thuc tren chi xay x^ = Tir cac diSu tren, suy diSu kien can cua Xo la nghiem cua phuong trinh da cho la XQ = hay x^ = Thu lai ta thay thoa man Vay phuong trinh da cho c6 nghiem la x = 0, x = b) Ta bien doi nhu sau: 3-(4- + 6- + 9-) = 25^ + • 16^ 12^ +18^ + 27^ = 25^ + • 16^ ^12^+14^-16^-25^=14^+16^-18^-27" 236 Gia sir la nghiem ciia phuong trinh, ta xet ham s6 fit) = t"> +(t + lyo - (/ + 4)^» - (/ +13)"" Taco /'(O = +2)^^-'-(/ + ) ' ° " ' + ) " ° - ' ) Phuong trinh da cho chinh la /(12) = /(14) va ro rang / ( / ) lien tuc tren (12; 14) nen theo dinh ly Lagrange, ton tai c e (12;14) cho / ' ( c ) = / ( l ^ ) ^ = 12-14 Do do, ta dugc Xo (c'"-' + (c + 2)"°-' - (c + 4)"°-' - (c +13)""-') = x,=0 y-' + (c + If"-' = (c + 4)""-' + (c +13)""-' C) ding thuc sau, d l thdy chi c6 the la A:^ - = <=> han JCQ = v i nSu khong, chSng - < thi c""-'>(c + 4)"°-',(c + 2)'"-'>(c + 13)"°-' ^ => c""'" + (c + 2)"°"' > (c + 4)"°"' + (c +13)"°"' Do do, XQ = 0,Xo = la dieu kien can de x^ la nghiem cua phuong trinh da cho Thu lai, ta thdy ca hai deu thoa man " Vay nghiem cua phuong trinh da cho la x = 0, x = Bai a) Xet ham s6 / ( x ) = x + e"' fix) = (l + x ' ) e"' - e" - e" [2 (l + 2x') / N2 Ta C O danh gia Suy fix) X tren M Ta c6: f\x) — , = + 2xe'' - e" va - f > => x' - X > - - ^ 2(1 + 2x' )e"'-" > le"'-" > 2e"\ > Vx e M Do fix) = c6 khong qua mot nghiem tren M Mat khac / ' ( ) = Hon nua x = la diem cue tieu cua ham so nen / ( x ) > / ( ) = Vay X = la nghiem nhat cua phuong trinh fix) = 237 b) Ta biSn d6i n h u sau: iii J US; t (;r + ) - ' - l ' - ' l - x = (jc + l) ^ - « (X +1)(S'-I^-'I -15^ -1|) + 5'"l'''l - X - = 5" <»(x + l)(5'"'^" 5^-1 - l ) = 5(5'-5'-IH) Ta xet cac truong hop: - Neu X < thi " - l | = l - % suyra 5'+- ^ - l ' - l = ' - < ' - ^ ' + ( ^ - l ) - l = ^ + ^ - = ( ^ - l ) va 5^-5'-l-'U5^_5i-(i-)^0 D o d o (x + l ) ( ' ' - l ) = 0«x = -l - NSu < X < t h i |5^ - l | = 5"^ - , suy 5HH _ _ ,| _ j ^ ' - ( i - ) _ (5- _ 1) _ ^ va D o do, truong hop nay, phuong trinh luon dung - N e u jc> t h i5 ^ - =5"-l,suyra ^ - - = 5'-'^-'^ - (5" - ) - = 5^-" - ' va 5" — 5'"l''~'l = — 5'+<'"^> = 5^ — 5^''' S u y r a (x + l ) ( ' - ' - S ' ^ ) = 5^-5'"^ « (jc + 2)(5'-^-5^) = « 5^"^ = 5" <^x = \, khong thoa man Vay phuong trinh da cho c6 nghiem la x = - hoac x e [0;1] l , ,v Bai6 a) Ta b i l n d6i n h u sau o (a + bf''"'" -a^"^"'" -b^''"'" = {a + by'°''' -a^""'' -b^""'' 238 ,2sin'jr 2sin'JT (_b_ a 1- a + b, a+b 1- = (a + b) , a Chu y rang < -N8u a+b sin^A:>- , b a+b (_b_ \a + b a + b, < nen xet cac truomg hcrp: thi 2sin^ x > l,2cos^ jc < vataco X 2sin^ JT \ sin X a ya + bj a a +bJ a a + b ya + bj < a +b a +b nen \^ jc a 1a+b a ^ ya + bj « +* = n0 a+b 2cos' X cos X Trong - >11 a + b, a Ka + b) < 0, khong thoa man - Neu sin ;c < - thi cung dan den mau thuan tuomg tu Do sin l-cos2x 2 X= - <» 71 , TT = -<^cos2x = Oo x = — + k—,k&Z b) Xet phuomg trinh 9' + = «• 3' cos(;rx) Ta CO bien doi 3' + 3^'" = acos(7rx) x^Xg (d Ta thay rang neu phuomg trinh c6 nghiem thi cung c6 nghiem la x = - jc^ Do do, dk phuomg trinh c6 nghiem nh4t thi ta phai c6 x,, = - <=> jc,, = Thay vao, tadugc 2-3 = acos(;r)<=>a = - Voi a = - , ta CO phuomg trinh 3" + 3^'" = -6cos(;rx) Theo bat ding thuc Cauchy thi 3' + 3^'" > I4T^ = 6, ngoai -6cos(;rx) < nen dang thuc xay Idii ;c = 1, thoa man dieu kien c6 nghiem nhat Vay gia tri can tim cua a la a = -6 239 Bai a) Phuang trinh da cho c6 the viet lai la X • 3^'-' + (x' -1) • 3^ +1 - X - jc' = « (x' -1)(3^ -1) + x(3^'-' -1) = Xet cac truong hop sau - Neu X = 0,x = ±1, ta thay cac gia tri deu thoa man nen phuang trinh tren CO cac nghiem la x = 0,x = ±1 - N I U X 9t 0, X 9t ±1, ta biSn doi tiep phuang trinh 3^—1 3^^' —1 +— = X -1 X 3'-l Vai /^0,taxethams6 sau / ( / ) = — e ~ ' ' R\{0} 3'-l De thay neu ? > thi 3'-1 > ^ / ( - — ^ > ; 3' - > 0, tuc la neu nguac lai / < thi ta ciing c6 3' - < => /(O = /(0>0,Vx^O Phuang trinh cuoi a tren chinh la / ( x ) + /(x^ -1) = nen de thay rang no v6 nghiem Vay phuang trinh da cho c6 dung ba nghiem la x = 0,x = ±1 b) Ta xet bien doi sau ^ Ta CO ^sin^ ;r _j_ ^ ^cos^ Jf ^sin^ x+cos^ J: — 2+ 2~^^ — 3.3^'"'^ + 3.3'=°''^ - 3''"'^«°''^ - = (3^'"'^ - 3)(3 - 3'°^'^) Do 0<sin'x,cos'x<l nen 3^'"'^-3 < 0,3-3^°^'^ > 0, suyra (3''"'^-3)(3-3'°''^)<0 Trong do, theo bat dang thuc Cauchy thi • 2" + 2"'^ = 2" + 2" + 2'^' > 3.^Ir^¥^T^ = 3, nen 2""' + 2"'" - > Dang thuc xay hai bat dang thuc tren va chi x = Vay phuang trinh da cho c6 nghiem nhit la x = 240 Bai a) Dieu kien xac dinh la x>0 B i t phuong trinh tuong duong voi 12^+16"-18^-32^ ^ 1-N/2 3\4"+4''-2\3'"-2'" 27" + 36" + 48" + 64" J2x " 3'" + 4".3'" + 3".4'" + 4'" 2^" (3" + 4") - 2" (3^" + 4^") ^ - V2 2'" - V2 2" (3"+4")(3"+4") <=> 2" ^ 2'" +- ^ > 3"+4" ^ 3^"+4^" V2^ 2' + -p,/>0.Tac6 Xethamso sau f(t) = — 3' +4' yft / T ^ +4^/•{()=- ' I n - + 2' ln2 / T n2 \ <0 + 2' .2, nen ham nghich biSn tren (0; + 0 ) Bat phuong trinh tren tuong duong voi fix) > f(2x) « jc < 2x K6t hop voi didu kien ta dugcbit phuong trinh da cho c6 nghiem la (0;+c») b) B i t phuong trinh da cho truong duong voi 3^"''^ + > • 3""' (*) Theo bat dang thuc Cauchy cho bon so duong, ta c6 3'"'-' + = 3'"'-' +1 +1 +1 > 4.3 > 4.3-1 ^ nen bat phuong trinh nghiem dung voi moi x c) Xet bat phuong trinh 2'-"-2x + l 2"-l > Dieu kien xac dinh x^i^O Ta thay rang ham so f(x) = - x + \ -2x +1 + ^ ham nghjch bien va / ( l ) = , / ( x ) > / ( l ) = O c i > j c < l « l - x > nen f(x) cungdiuvoi \ Ta cung thay rang ham so g(x) = 2" - la ham d6ng bien va g(0) = nen g(0) > o X > nen g{x) cung dau voi x 241 Suy bat phuong trinh da cho tuomg duomg vod -—- > <=> < x < X Vay tap nghiem cua BPT la (0;1] Bai9 a) Xet he bat phuomg trinh ^4024 ^^20.2^2012^^+2012^ x^°^^+x^°"<20lP+20ir Dat 2012 = ta xet bai tong quat: jx'^-'+x^-'<(>;-l)'^+(>;-l)'^" Bat phuomg trinh thu nhdt tuomg duomg vol x^' - + - > » (x' - y" )(x^ + / ) + x^ - jv" > <=>ix''-y'')ix'+y''+\)>0 D I thiy x^ + ^''^ +1 > nen ta duac x^ > y' Tathay x^ > y « • >^lnx > xln>'<=> >— X >' > 2012 X Ta lai dat 2011 = z , bit phuomg trinh thu hai tuomg duomg vai x'^+x'<z''+z\ ^i , ^ Inx ln2011 Bien doi tuomg t u nhu tren, ta duac — < ^ • X 2011 Do do, he bdt phuomg trinh da cho tuomg duomg vai hi2012 Inx In 2011 2012 ^ X 2011 ^ Xet ham s6 / ( / ) = ^^,t > Ta thiy r i n g /'(O = < 0, > Suy ham s6 nghich bien tren (3, +QO) ^ In2012 Inx ln2011 Dodo, < < « 2011 < x < 2012 2012 X 2011 Day chinh la nghiem ciia he bat phuomg trinh da cho 242 x>0 b) Xet phuong trinh > jc Di6u kien: 1- 2x-\ Bk phuomg trinh da cho tuomg duong voi 6-y>-^ol-y>-A_ 2x-\ • Voi x> — ta c6: - S'' < < —-— nen bat phuomg trinh v6 nghiem 2x-\ • V a i < X < ^ Xet ham s6 f{x) = 1-3' tren Suy ham s6 nghich biSn tren khoang Xet ham s6 g{x) = — ^ tren 2x-l CO Suy ham so nghich bien tren khoang CO fix) Dodo g'ix) = - 10 (2x-l)^ ^-3' f{x)>f ln3<0 = 1-V3 <0 D o d o ^ W < g ( ) = -5 Tir suy / ( x ) > - V3 > -5 > g(x) Do do, thi bat phuomg trinh nghiem dung Tu cac truomg hop tren, ta c6 nghiem cua bdt phuomg trinh da cho la ^ 1^ v''2, Bai 10 a) Ta b i l n d6i bk phuomg trinh nhu sau 3^'-" - + (x' - 4)3''"^ > Ta xet cac truomg hop sau: -Neu x>2 hoac x<-2 thi " - - > ^ 3^'-' - > 0,(jc' - 4)3''-' > nen b i t phuomg trinh nghiem dung -NSu - < x < thi x ' - < ^ ' ' " ' - < , ( x ' - ) ' ' ' ' < nen bat phuomg trinh khong thoa 243 Do do, nghiem cua bat phuomg trinh da cho la (-00; -2] u [2; +c3o) b) Xet bat phuong trinh a""'^" > cos^ jc, VJC e M DiSu Icien < o ^ Dat / = 2cos^ X => < / < Yeu cau bai toan tuong duong vai viec tim so thuc a > cho a'-' > ^ V ^ e [ ; ] Ta thay rang vai t = Q thi bat dang thuc dung va khong phu thuoc vao a nen ta chi can xet Q<t<2 Lay logarit nepe hai ve cua bat dang thuc, ta c6 - ( r - l ) l n a > l n / , V / e ( ; ] (*) De thay bat dSng thuc cGng luon dung vai t = nen ta chi can xet hai khoang sau: - Neu r ( ; l ) thi / - l < , l n ^ < , (*) tuomg duang vai \na<y^ thuc phai dung vai moi t e (0;1) nen In a < lim Bat dSng = -Neu r e ( l ; ) thi / - l > , l n ? > , (*) tuomgduomg vai \na>j^ B i t ding thuc phai dung vai moi t e (0; 1) nen In a > lim = Do do, ta can phai c6 In a = « a = e de thoa man ca hai bat ding thuc tren Thu lai, vai a = e, ta can chung minh / - l > l n / , V / e ( ; ] Xet ham s6 /(M) = M-l-lnM,w6(0;2] Ta CO f'(u) = - — = - — - , / ' ( « ) = ^ > u u u cua f'(u) = hay cue tieu cua ham / va ^ - > In f, nen f{u) dat cue tieu tai nghiem la u = l nen / ( M ) > / ( l ) = l - l - l n l = e (0; 2] la dung Vay gia tri c i n tim la a = e 244 ChUcfng P H l / d N G TRINH, BAT PHLfdNG T R I N H L O G A R I T A Tom t ^ t ly t h u y l t 1) Cdc cong thuc Men doi dai so: X • log„x + log^>' = log„xy,log„x-log^>' = l o g „ - vai a,6,ceRM{l} y • log> = — = log* a log l = 0,log * • ^ i ^ ^ = log„clog,6 vai a,*,ceRM{l} log, b = - l o g , vai a,beR^\{\},a^0,/3eR a 2) Cdc ngi dung ve gidi tich: Ham s6 luy thira y = log^ x vai < a la R i thi tap xac dinh la va tap gia tri - i - I - Dao ham cua cac ham so (log^ x) = —^-— xlna Tinh dan dieu ciia ham so: • Neu a> \i cac ham so >' = log„ x dong bien • NSu < a < thi cac ham s6 >'= log^ X nghich bien Giai han cua ham so: " • Vai a>\, ta c6 cac giai han sau lim log^ x = +co, lim a' = - c o • .^ Voi < a < ta CO cac giai han sau lim log^ x = -oo, lim a"" = +oo, Tilp theo la cac dang phuang trinh ea ban: • Phuang trinh log„ x = m luon c6 nghiem nhat la x = a"" • • Phuang trinh log„ / ( x ) = log„ g(x) / ( x ) = g(x), cung tinh don dieu Phuang trinh log^ / ( x ) = log^, g(x) vai a ;^ * thi c6 cac truong hap sau: - NSu (a - \)(b -1) < thi hai ve cua phuang trinh c6 tinh dan dieu khac nhau, phuang trinh c6 khong qua nghiem va ta can nham true tiep nghiem 245 - Neu a^b \a (a-\)(b-\) >0 thi ca hai vg dhu c6 cung tinh dan dieu ta dat an phu (dat ca hai ve dat cung mot biSn mai) dl mu hoa hai ve va chuyen ve mot phuang trinh mu khac Day la cac dang ca ban va noi chung hau het cac phuang trinh mu va logarit tu ca ban den nang cao deu phai thong qua Trong goc nhin nay, mot s6 dang bign nhu phat hien tinh d6i xung dk chia va dat in phu, hikn d6i ap dung cong thuc thich hap de dua ve ciing ca so, Day la cac bai toan ve mu va logarit thuang xuat hien cac ky thi Dai hoc Ben canh do, ta ciing c6 the giai bang each su dung danh gia bSng cac hit dSng thuc quen thuoc hoac su dung tinh don dieu ciia ham s6 thong qua viec khao sat true tiep hoac dung dinh ly RoUe va Lagrange Cac bai lien quan dSn giai tich nhu thi thuang kho horn va doi nhiSu danh gia phuc tap hon - Vai < a 9i va Of > thi log„ x cung dku vai {a - l)(jc -1) -Vai < a ^ l va jc,>'>0 thi log^x-log^>' cungdauvai (a-\){x-y) -Vai 0<a,b^\0 thi log^jc-log^x cung dau vai x{a-b) B P h i p c n g p h a p giai Phu-cmg phap bien doi ve cung ca so Nhan xet: Tuang tu vai phuang trinh mu, phuang trinh logarit cung c6 cdc dang bien doi true tiep dua ve ciing ca so de suy hai doi so tuang ung bang Vi du Giai phuang trinh + log,(x + ylx^-\) = log,(x + V x ' - l ) log^(JC-ylx'-\ L&igiai *) DiSu kien xac dinh la x>\ Dat / = jc-Vx^ -1 o Ta c6 X + yjx^ - = - Do log21 + log3 j- = log, y « logj / - logj / + log, t = <=> log^ t{\ log3 + log, 2) = <:> / = Ta CO phuang trinh x-yjx^-\ yjx^ -I <=>x^-2x + l = x ^ - l o x = l, thoa man di8u kien da cho Vay phuang trinh da cho c6 nghiem nhit la x = \ Vi du Giai phuang trinh logjClogjOog^ x)) = log4(log3(log2 x)) L&i giai *) Dieu kien: log3(log4 x)>0; log3(log2 JC) > 246 Phuong trinh da cho tuong duong voi 10g4 0og3(log4 X)f = I0g4(l0g3(l0g2 X)) » (I0g3(l0g4 X)f = lOgjOog^ X) o (log, (log, x)f = log3(21og4 X ) « (log3(log, x)f = log3 + log3(Iog4 x) Dat t = log3(log4 x) > Ta CO phuong trinh bac hai bien t la • , + J l + 41og,2 =log32 + / « / = — ^ ^ , d o />0 Khi do, ta CO x = 4^', , , ,.3 Vi du Giai phuong trinh - log, (x + 2)^ + = log, (4 - x)^ + log, (x + 6) *) Dieu kien - < x < 4,x ?t - LMgidL Phuong trinh da cho bien doi i log, ( X + 2)^ +1 = i log, (4 - X)' + i log, ( X + 6)' o log, (4|x + 2|) = log, [(4 - x)(x + 6) « | x + 2| = ( - x ) ( x + 6) (*) -N6u X > -2 thi phuong trinh (*) tuong duong voi 4(x + 2) = ( - x ) ( x + ) O x ' + x - = 0c:>(x-2)(x + 8) = « > x = -NSu X < -2 thi phuong trinh (*) tuong duong voi -4(x + 2) = ( - x ) ( x + ) < » x ' - x - = < » ( x - l ) ' =33<=>x = l->/33 Vay phuong trinh da cho c6 hai nghiem la x = v x = l - >/33 Vi du Voi 0<a^\, hay giai va bien luan bat phuong trinh x'°^»^-a'>0 L&i giai *) DiSu kien x > Taco x'°^°^-a'>0«a'°«"^'°«"^-o'>0<:>a"'^°^-fl'>0 Theo nhu nhan xet o tren thi a'' - a ^ cung dau voi {a - l)(x - y) voi moi x,>'e]R,dod6 a'ozU _a'>Q^(^a-l)(log^ x - ) > O ( o - l ) ( l o g , x - V2)(log„x + 2) > Ta lai c6 log^ x - log„ y cung din voi (a - l)(x - >') voi moi x, € IR, do (a-l)(log,x-V2)(log,x + V2)>0 » ( a - l)(log„ X - log„ )(log, x + log„ ) > « ( a - ) ' (x - )(x - ^ ) ^0 Ta cung thay rkig — ^ > <=> a - > Ta c6 cac truomg hop sau - Neu a = \i bat phuong trinh luon dung va nghiem cua no la (0;+co) - Neu < a < thi nghiem cua b i t phuong trinh la [a^;a~^] - Neu a > thi nghiem cua bat phuong trinh la (0; a""^ ] u {1} u [a^; +oo) V i du Giai phuong trinh sau log ( l • 5"^ +15 • 20"^) = ;c + log 25 L&igidL *) Dieu kien: x > Phuong trinh da cho tuong duong voi: 10-5^+15-20^ =25-10^ » +3-2'^ =5-2^ o 3-2'^ - - ^ + = 2' = ;c = <=> X = logj 2^ = 3 x =0 x = l-log23' Vay phuong trinh da cho c6 hai nghiem la jc = 0, jc = - logj V i du Giai phuong trinh logj (x^ - 5x + 4) + logj (x - 4) = - log, (5x - 5) L&i gidL *) Dieu kien x > Phuong trinh da cho tuong duong voi ^- > logj ( x ' - 5x + 4) + log, (x - 4) = - logj ' « log, P'^^ 5(x - ) 5x-5 + log, (X - 4) log, = « log, (x - 4) - log, + log, (x - 4) log, = log, (x - 4) = l + log^S + log5 Chu y rang l + log5 = logjlO = log^ nen x = - ' ^ -f Vay phuong trinh da cho c6 nghiem la x = 2'°^'^ + V i du Giai bat phuong trinh logs x + x - x^^4 >1 L&igidL *) Dieu kien < x ^ Bat phuong trinh da cho tuong duong vol log3X + x - _ ^ ^ Q ^ l o g X - ^ Q ^ x-4 x-4 248 Ta xet tnxomg hop: jc>25 jc>25 jc>4 x<25 - Neu l o g X - < <=>x<4 x-4<0 x<4 K€t hop lai, ta ttidy bit piiuomg trinh da cho c6 nghiem la x G (0;4) U (25;+OO) ' Bai tap phan Bai Giai phuong trinh logj x + \og^ x + log^ x = log, x Bai Giai phuong trinh log27 (logj x) = logj (logj, ^) + ^ • Bai Giai phuong trinh sau log, + log , - = l o g ^ (125x + 6) Bai Giai phuong trinh (4' - 2^^' - 3) logj x - = ^ -4" X X X Bai Giai phuong trinh log^ -2 \og^°^3- log, - = logj x + log3 x + log, x - Bai Tim dieu kien cua a de ton tai x thoa man he didu kien X y-4>5' l + log2(a-x)>log2(xVl) Bai Giai phuong trinh log^ (x - Vx^ -1) log, (x + Vx^ -1) = log^o (x + Vx^ - ) Huong dan giai bai tap phan 1 Dieu kien x > Ta c6 bien doi sau logj X + logj X + log4 X = log, X o logj • log, X + log3 • log, X + log4 • log, x = log, x log, x(log, + logj + log4 -1) = « log, X = <=> X = 1, logj + logj + log4 - ?t Vay phuong trinh da cho c6 nghiem la x = Xet phuong trinh logj^ (logj x) = logj (logj^ x)+^ Dieu kien x > Ta c6 249 10g27(log3^) = 10g3(lOg27^) + 1 f\ \ -<=>-10g3(lOg3x) = log3 - 10g3 X + - 1 « - logj (log3 x) + log3 + = ^ logs (log3 x) = « l o g X = o X = 27 Vay phuong trinh c6 nghiem la x = 27 Xet phuong trinh logj x' + log, - = l o g ^ (125x + 6) Dieu kien x > Ta c6 log3 x^ + log^„ = log (125x + 6) « log, (8x^) = log, ^{USx + ef 53 « 8x' = ^(125x + 6)' » lex" = 125x + 2x = ^125x4-6 (x - 2)(16x' + 32x' + 64x + 3) = "x = 16x'+32x'+64x + = V i X > nen phuong trinh sau v6 nghiem Vay phuomg trinh da cho c6 nghiem nhdt la x = DiSu kien: x > Ta c6 cac bien doi sau 4- _ 2"^' - = ( " - • 2^ - = (2^ +1)(2" - 3) va x+\_ " - ^ + = 2-2^-4^+3 = - ( ^ + l ) ( ^ - ) x+l Suyra ( " - " " ' - ) l o g X - = - " 2^=3 10g2 X = -1 « ( " +1)(2^ - 3)(log2 X +1) = X = logj X=— Vay phuong trinh da cho c6 hai nghiem la x = logj 3,x = ^ X X X Xet phuong trinh logj - logj - log, - = logj x + logj x + log, x - DiSu kien x > Phuong trinh da cho viet lai duoi dang (log2 X - ) (log3 X -1) (log, X - ) = log2 X + log3 X + log, X - o log2 X log3 X log, X = log2 X logj X + log3 X log, X+ log, X logj X Ta thiy x = thoa man phuomg trinh Xet x ^-^ thi phuomg trinh tren tucmg duomg voi —?— + — ! — + — ! — = logjX logjX log, + log, + log, = « log, 30 = « X = 30 logjX 250 ' Vay phuang trinh da cho c6 nghiem la x = 1, x = 30 3"-4>52 Xet he bit diSu kien • l + log2(a-x)>log2(x^+l) X X Taco ' - > < ^ ' ' - ^ - > X X Datfix) = 3'-52 - Ta thiy phuomg trinh ^ - - = 0<::>l = rv5 + - CO nghiem nhat la x = nen nghiem cua bdt phuong trinh thur nhdt la [2;+00) » a - 11 Bat phuang trinh thu hai tuang duong vai logj2(a-x)]>log2(x'+l) ^ 2(a-x)>x'+\ a>^ + x + -^ He CO nghiem va chi (*) c6 nghiem thuoc (*) [2;+00) x" Dat g(x) = Y + x + - Ta c6 g'(x) = 2x^+1 > 0,Vx > nen g(x) d6ng bien 21 tren [2;+co) chi a > va ta cung c6 g(2) = — nen (*) c6 nghiem thuoc [2;+00) va 21 Vay dieu kien can tim la a > 21 Dieu kien x > Ta c6 bien doi nhu sau log,(X - Vx^-1)log3(X + V x ^ - ) = log,o • log,(x + V x ' - l ) log5(x + Vx' - ) = <=> l o g ^ C x - V x ' - I ) = log2o5 X + Vx' -1 = x-Vx'-l =4'°«»' Phuang trinh thu nhat cho ta nghiem x = Tir phuang trinh thu hai, suy x + Vx^ -1 = 4"'°^'°^ Do x =—(4'°^™^+4"'°^^°'^ Phuang phap dat an phu dua ve dang dai so Trong cdc phucmg trinh logarit, de qua trinh bien doi sang sua horn, ta c6 the dat cdc bieu thicc xudt Men nhieu Idn lam an phu roi xu ly phucmg trinh dan gian han da thu duac Trong vdi truang hap, ta cdn phai ket hap them mot so bien doi nhdt dinh maiphdt Men duac bieu thicc cdn de dat dnphu 251 Vi du Giai phuong trinh sau 2'°^'^' _ i o g ( x ^ - ) ^3.4iog,(.-2) ^ Q L&igiaL *) Dieu Icien x > Ta bien doi nhu sau: 221og3Ar _-^,^+\o%-iX+\og^(x-2) <=> ^ \ o z , x ^ ^ _^_2^o%,x _|_ g _ 2^og-,(x-2) y<'g3(^-2) 2log3(j:-2) = 2"'«'^-'°«'<^-^' = _Q _^^^2^o%y(x-2)^^ Chia vg cho (2'°^''^-^>)' > 0, ta dugc /= /^-6/ + = 0c^(^-2)(^-4) = 0<=> Neu / = thi _Q -6- , , +8 = 2log3(Jt-2) > 0, ta CO t =2 t=4 = » logj — ^ = <=> — — = « jc = x-2 jc-2 x-2 Vay phuong trinh da cho c6 nghiem la x = 3,x = — Vi du Giai bat phuong trinh sau 2""^' + L&igidL *) DiSu kien xac dinh < x ;t — log^/;A Dat 2^ - 20 < - =^>0 t h i / ' + r - < < » ( / + 5)(?-4)<0 h a y - < / < Suyra/<4<»22 < « - l o g ^ x < <^ logj^ x < Dodo ( V ) " ' < x < ( V ) ' « ^ < x < - -2 < log^ x < ^ Vay nghiem cua bat phuong trinh da cho la — < x < 2,x ^ ' • Vi du Giai bat phuong trinh log, (4^ + ) > log, (2'^"' - • 2^) 2 (DH khSi A 2002) Lo^^/di *) Dieu kien: 2'^^'-3-2^ > « 2^*'>3 <=>x>log2 - ,, Xet bat phuong trinh log, (4^ + ) > log, (2'""' - • 2^) 252 Taco ^ + < ' " " ' - - ^ « ^ - - ^ - > V Dat/ = 2" > thi r ' - / - > « ( / + l ) ( / - ) > < i > r > Suy 2"^ > <=> X > Do do, nghiem cua bit phuong trinh da cho la A: > Vi du Giai phuong trinh sau 4'°^^'" -x'°^'' = 2.3'°^^'^' Lo"/^/di *) Di^ukien jc>0 Ta biSn d6i nhu sau: ^l+logjJT _ g l o g J C _ 2.32+21082;: 2lOB2^^^ 3l0g2-' 2lOg2 Dat r = 31082 = 18 /2^'°82^ > 0, ta CO 31082 / ' - r = « ( ^ + 2)(4/-9) = 0<:*/ = ^ (2 Nlog2 , \ = - <=> log, X = -2 X =- 4 Vay phuong trinh da cho c6 nghiem la x = - Vi du Giai phuong trinh sau log^Cx-l)^ +log4(x-l)^ =25 Do L&igidL *) DiSu kien x > Ta bien doi nhu sau log: (x - If + log] (x - ) ' = 25 « log: ix-l) + log] (x -1) = 25 Dat t = log^Cx-l) > 0, ta CO 16/'+9r = « ( / - l ) ( / + 25) = » / = l Do do, ta dugc log^ (x -1) = <=> log4(x-l) = l log,(x-l) = - l "x = <=> • x=— Vay phuong trinh da cho c6 nghiem la x = 5, x = — Vi du Giai phucmg trinh sau logj^ x + log^^, x = ^ L&igidL *) Dieukien x>0,x^—,x^ —7= yj2 I 253 Ta bien doi nhu sau: logjX log2 2xj H log, X — = — •» l + logjxj +- logjX l + 31og2X log^ 2x' Dat / = logj X thi /'(I+ 30+ /(! + /) +\t + 3/ = —2 <;:> (i+30(i+/r — Thu gon va phan tich nhan tu, ta ducrc (/-l)(5/^+4/ + l) = o t = l 5/^+4/ + l = 0" Do do, phuong trinh thu hai v6 nghiem logj x = <» x = Vay phuong trinh da cho c6 nghiem la x = 2x Vi du Giai b4t phuong trinh sau log^ - x -4<V5 V 2V LatigidL *) DiSu kien xac dinh: 2x >0<»0<x<2 2-x 2x_ = log^ 2x ^ nen bat phuong trinh tuong duong: 2-x Vi log log2 2x - Iog2 2x - + 2-x 2-x Dat t = log 2x thi ta dua ve 2-x ( / - ) ( / + ) > -3 < / < -2 <»0<(/-2)(/ + 2)< VS « [(/-3)(/ + ) < 2</<3 2x l < ^ < i — < x < -3<Iog3 - x <-2 2-x4 17 2x < ^ < < log, <3 x < x < x L3 So sanh voi dieu kien ban dau thay thoa man ^ 2'lu (A 8^ Vay tap nghiem ciia bat phuong trinh la 35 U7'9 254 Baitapphan2 -iog hog X X Bai Giai phuong trinh 2x^°^''' =2^ "^'"^ Bai Giai phuong trinh log, (3" +1) • logj (3"^^ + 9) = 24 Bai Giai phuong trinh log2^—(log2 Bai Giai phuong trinh log3^^7 x) +(log2x) =1 (9 +12x + 4JC^ ) + log^,^, (6x^ + 23x + 2l) = (DHKinhtiqudc Bai Giai phuong trinh log^^^^^^,^, x + log^.^^^^^^^x' = l o g , , , , , , ) X Bai Giai bdt phuong trinh 16 log^^^, x - log3^ jc^ < dan 2001) (Du bi khSi A 2002) Bai Tim nt de phuong trinh sau c6 nghiem thuoc (0;1): A{\O%.^4X^ -log,x + w = (Du bi khdi B 2003) Bai Giai phuong trinh 2(log2 x +1) log^ x + log^ =0 (Du bi khdi D 2006) Huong dan giai bai tap phan Dieu kien < x ?t Ta lay logarit hai 2x^'"" log2 V •o log2 theo co s6 , ta c6: = 10g2 J \ + — log2 X • log2 ^ ^ <=> + logj Dat y = logj X thi ta dugrc + y =3y^iy-\)(y-2) X = log2 x j y= \ = 0<^ y-2 l0g2 X = logj X = x = x = 4' Vay phuong trinh da cho co nghiem la x = 2,x = Ta bi6n d6i nhu sau: log3 (3^ +1)• log3 (9(3^ +1)) = 24 « log3 (3^ +1)(2 + log3(3^ +1)) = 24 Dat y = logj^i" +1) > thi ta co phuong trinh y(2 + y) = 24<:>{y-4)(y + 6) = 0<^ y=^4 255 Khi do, ta dugc logjCS" +1) = « 3^ +1 = 3' « 3" = 80 » X = log, 80 Vay phuong trinh da cho c6 nghiem la x = logj 80 2 Xet phuong trinh logj^ -(log2 x)^ +(log2 x)^ = Di6u kien < x - log Ta CO log2, - = X ^ = Dat log2 2x + logj X \y c>{\-y)[(\ 72 Voi yv == — ,tac6 Vai \, taco x =x2'= 2=2.^ = = log2 x thi dua vg = 0« y=\ yf-/ Vay phuomg trinh da cho c6 nghiem la x = 2, x = Digukien 3x + >0,2x + > , x - Phuong trinh da cho tuong duomg voi log3,,, (2x + 3)^ + log2,,3 [(2x + 3)(3x + 7)] = « log3,,, (2x + 3) + log2,,3 (3x 4- 7) = Dat t = log3^+7 (2x + 3), ta dugc 2? + - = 3«2/ - / + l = 0<:>r = l v / = 1- / -Voi / = thi log3^^7(2x + 3) = l o x + = 2x + 3<=>x = - , loai \^ -Voi t = ^ thi log3^,7 (2x + 3) = ^ « V3x + = 2x + <=> (x + 2)(4x +1) = Phuomg trinh c6 hai nghiem phan biet la x = - v x = —^, ta th§y chi c6 X = - — thoa man nen phuong trinh da cho c6 nghiem nhit la x = - — 4 256 Dieu kien x>0,x(24x + \f ^l,x\24x + \)^\ Tnrac h6t, ta thdy x = thoa man phuang trinh Xet x^\, nhu sau: 1 log, x(24x + \y -+- log^, ta c6 the b i l n d6i (24x +1) log, (24x +1) 1 <=> l + 21og,(24x + l) + i^^iog^(24x + l) Dat / = log, (24x +1) thi j — ^ + log.(24x + l) - = - Quy dong va rut gon, ta dugc r=l ? ^ - / - = « > ( / - l ) ( / + 2) = « > ~ Vai / = l , t a c log,(24x + l) = » 24x + l = x x = — ^ < , khong thoa man 2 Vai / = — , t a c log,(24x + l) = — « - x ^ =24x + l Dat y = x^ > , ta c6 3 \ / +1 <^ (2>'-1)(12y* +6y' +3y' + 2y + l) = 0<:> y = ^ ^ 1 Khi do, ta CO x^ = - <=> X = - , thay vao dieu kien xac dinh, ta thay thoa man Vay phuong trinh da cho c6 hai nghiem la x = - , x = ^ >••>•', ^:-, , Xet bat phuang trinh 16 log^^^j x - logj, x^ < x>0 Dieu kien Ta bien doi nhu sau 3V3 3' 16 3^"^^^ l o g 27x^ Dat / = l o g , ' X .u^ thi 8/ cOo l o g 3x 3/ 2/ + / + ^ < <=> ^^^^jX 31og3X 21og3X + 2t^-t (2/ + 3)(r + l ) l o g X + l ^ t(2t-\) <0« — — < (2/ + 3)(/ + l ) Lap bang xet dau cho bat phuang trinh, ta thu dugc < / < - hoac < / < ^ 257 Suy -— < l o g , x < - o < x < - hoac < l o g , x < — ci>\<x<yl3 ' 3V3 • ' Vay nghiem cua bat phuong trinh da cho la x e 3V3'3 Didu kien x > Ta biSn doi phuang trinh da cho ( l o g x ) ' + l o g X + w = Dat/ = l o g X thi t^+t + m = Phuong trinh CO nghiem thuoc (0;1) / ( ) / ( l ) < vori / ( / ) = + r + w hay m{2 + /n) < « -2 < m < Vay di8u kien can tim la -2 < w < Phuomg trinh da cho tuong duong voi (logj x + l ) l o g x - = ••••si J ^ - ^ Di8u kien x > Dat / = log^ x thi <=> t^-2 log2 X = log2 X = -2 <=> x =2 1• X = — Vay phuomg trinh da cho c6 nghiem la x = 2, x = — >i \ Phuang phap dat an phu dua ve dang mu Nhan xet: Nhu phdn ly thuyet ban ddu, mu hoa la mot cdc phuomg phdp di gidi quyet phuang trinh logarit truang hap hai ve c6 ca s6 khong bdng vd deu cung Ian hodc ciing be hem Ta dat an phu mod bdng gid tri cua hai vi cua phuang trinh vd chuyen phuang trinh logarit da cho vi phuang trinh mu, dang de xif ly han , V i du Giai phuomg trinh sau log3 (x +1) = log2 (x^ + x ) L&igidL *) Dieu kien x+ l>0 <::>X>0 " x(x + ) > Ta CO logj (x +1) = logj (x^ + 2x +1) Dat log2(x^+2x + l) = l o g ( x ' + x ) = / thi x^+2x + l = ' , x ' + x = 2' (2^ f Do ' + = 3' « — + — = D I thay ve trai la ham nghich bien nen phuomg trinh c6 khong qua mot nghiem Ta thay t = thoa man 258 K.k hop diku kien ta dugc nghiem x = -1 + Vs •/ Khido x'+2x + l = 3<=>x^+2x-2 = 0«x = -l±V3 Vi du Giai phuong trinh sau log,(log;i^|x^+\ = log, (log, (Vx^+1 + x)) 5 L&i gidL *) Ta thiy Vx^ +1 + x > 0, Vx^ +1 - x > nen phuong trinh da cho xac dinh tren E Ta CO I V ? + l + X JI V ? + l - ^) = nen phuong trinh da cho c6 th8 \ik lai la ^ l0g3(l0g5(V^-x)) = l g , ( l g ( V ^ - x ) ) ! U v i : G Dat a = log; (Vx^ +1 - x) Ta c6 log3 a = log, a Ro rang tinh don dieu cua hai ve khaclog; nen phuong khong nghiem, thiy a== -1—thoa (Vx^n - x) = 1trinhVx^c6 +1 - X =qua <=>mot Vx^n = x +ta5<»x nen Thu lai ta thay thoa man Vay phuong trinh da cho c6 nghiem la x =^ Vi du Giai phuong trinh sau \og^ cot x = logj cos x -,:,, ^ r, L&igidL *) Digu kien cosx > O,cotx > <=> x e {k27r,^ + k27r),k e Z Dat logj cot X = logj cos x = r Ta c6 12 cot X = 32,cos X = 2' => cot^ X = 3', cos^ X = 4' J nit r - n ^- u '4(i' cos^a COS^fl ^ , Mat khac cot a = — — = Va k^r nen 4' •<:>3' =12'+4' «4' + v3 = (*) 3'sin= a 1-cos a 1-4' Ve trai la ham tang theo biln / nen phuctng trinh c6 khong qua mot nghiem Ta lai thay / = -1 thoa man phuong trinh nen (*) c6 nghiem nhat la / = - l Tu r = -l,suyra cosx = —:^x = —+ A:2;z-,A:eZ (do dieu kien xac dinh diu) Vay phuong trinh da cho c6 nghiem la ^ = y + k2^,k e Z A y r - ^ i U ' 259 Vi du Cho x,y la cac s6 thuc duong thoa man di^u kien log<,x = log,>; = log,(x + >') Tinh ti s6 — y x = log^ y = log^ (x + y) = t L&i giau *) Dat logg Suyra 9' =x,6' = y,4' =x + y hay 9'+6' =4' « Dat « = ^3^ \1j >0 thi a ^ + a - l = 0<=>a = -1±V5 42'- =0 nhung a > nen a = -i+Vs X 9' 3' ^ ^ X - l + Vs Hon nua — = — = — = a nen ta duoc — = y 6' 2' - y Vi du Giai phuang trinh log^^^ (JC' - 2X -11) = l o g ^ ^ (x' - 2x -12) L&i gidL *) D i k kien - 2x -12 > Ta thay ring (2 + V5)' =9 + 4V5, (2^2+7?)' = + 4^5 nen c6 thg biSn doi phuang trinh tren ilog^^^(x^-2x-ll) = l l o g ^ ^ ( x ^ - x - ) « log9.47i(^' - 2x -11) = log,,,^(x^ - 2x -12) Dat a = + 4Vs thi log„^, (x^ - 2x -11) = log^ {x^ - 2x -12), ti6p tuc dat log,,, ix' - 2x -11) = log, (x^ - 2x -12) = / «(a + l ) ' = x ^ - x - l l , f l ' = x ^ - x - Suy (a +1)' =a' + \ Den day, ap dung bit dang thuc Bernoulli, ta c6 t = \ hay x ' - x - = + 4V5 «x = + 2V5 v x = -2V5 De thay hai nghiem thoa man diSu kien dh bai nen phuang trinh da cho c6 hai nghiem la X = + 2>/5,x =-2V5 Vi du Giai phuang trinh log5_^(log2 x) = log7_^(log2 2x) L&i gidL *) Dieu kien < x < , X t Truac het, ta thSy day la dang phuang trinh c6 dang hai biSu thuc logarit b§ng 260 nhung khac ca s6 Do do, ta se dat §n phu truac dk don gian hoa biSu thuc Dat log,_,(log2 x) = log7_,(log2 2x) = t Suy .1 log2X = ( - x ) ' f logj X = (6 - x)' log2 2JC = (7 - x ) ' [1 + log2 X = (7 - x)' Tiep tuc dat - x = o > Ta c6 phuong trinh fl'+l = (a + l)' « a +1 + a +1 = 1 Do < a < nen ta c6 the xet cac truong hop sau a+1 a+1 - NSu / > thi Y>—^+ a+1 a - Neu / < thi a+1 + a +1 - Neu r = thi a + a+1 a+1 < a + 1) a a+1 a+1 + = 1, loai , = 1, loai a+1 " + ^ — = l,th6aman a+1 a + Do do, / = <=> X = Thu lai thay thoa man Vay phuong trinh da cho c6 nghiem la x = Bai tap phan Bai Giai phuong trinh log^ x = log, (\/x + j Bai Giai phuong trinh logj (logj x) = logj (logj x) Bai Giai phuong trinh log, (s" +1) = log24 (25" - ) Bai Giai phuong trinh (3 + ^t^*' + x(3 - ^p*' = x^ +1 Bai Chung minh rang phuong trinh log, (>/x + 2Vx j = log4 x c6 nghiem x^ thoa man cos ^ > 2014 sin 4xo + V2 Bai Giai phuong trinh sau log; + V3'' + l ] = log4 (3'' +1) Bai Giai phuong trinh sau 3'°^^' = x^ - 261 H u o n g dan giai bai tap phan • ^ = ^ ^'' Xet phucmg trinh log^ X =^ logj (\/jc + 2^ Di^u kien X > Dat log, X = logj (Vx + 2) = / thi x = 7' = 3' nen suy V^ + (V7)'+2 = 3' v3 +2 = D I thiy trai la ham nghich bien theo / nen phuong trinh c6 khong qua nghiem Thu true tiSp, ta thiy t = thoa man Khi x = 49 Vay phuong trinh da cho c6 nghiem nhit la x = 49 Xet phuong trinh sau log2 (logj x) = log3 (log2 x) Di^u kien x > Dat log2 (logj x) = logj (log2 x) = •log3X = 2' , Y t^ log2X = 3'" Chia tung vS hai ding thuc, ta dugc log3 2= - <=>r = log2 log3 Do do, ta tinh dugc x = 3^' voi / xac dinh nhu tren Xet phuong trinh log^ (s' +1) = log24 (25" - ) DiSu kien x > Dat log, (5" +1) = log24 (25" -1) = / thi > + l = 6' Suy 25"-1 = 24' (6' - ) ' = 24' +1 36' = 24' + • 6' « '2^ +2 .3; = D I thay vl trai nghich bien nen phuong trinh c6 khong qua nghiem Thu true tiep, ta thay t = thoa man Khi x = Vay phuong trinh da cho c6 nghiem nhat la x = Xet phuong trinh (3 + >/5)'°^^" + x(3 - V5)'°«^" = x' +1 Qiku kien x > Chu y rang (3 + >/5)(3 - Vs) = nen (3 + Vs)'°^*" • (3 - VS)'°^^" = 4'°^^" « (3 + yfsp*" • (3 - VS)'°^^" = x Dat (3 + V5)'°^^" = J > thi ta c6 (3 -yfst^'" = ^ 262 Ta CO phuomg trinh ^' , / y 2 y N6u = ! thi (3 + ^)'°^" =\<^\og,x = 0<=>x = \ N6u = thi (3 + y/sf"^*" = tinh logarit ca s6 cua hai ve, ta dugrc , log^ X logics + VS) = log^ X » l o g ^ X [logics + x/5)-2] = ^ x = l Vay phuong trinh da cho c6 nghiem la x = Xet phuong trinh 21ogg (A/X + 2ifx^ = \og^ x Di6u kien xac dinh x > Ta biSn d6i duoc log^ (Vx + 2Vx) = logj V x Dat log,(V^ + 2V^) = l o g V ^ = / thi V^ + 2V^ = 6',V^ 4'+2-2' =6' « .3 + 2- = 2',suyra = Ve trai la ham so giam theo / nen no c6 khong qua nghifm Dat/(0 = /2V + 3- -1 thi /(l) = | + ^ - l > , / r3^ .2j va ham / ( / ) lien tuc tren mien mien Do do, f(t) Ta CO — e = (2^ .3; +2 rn n X 3>/3 -1<0 1;— nen no c6 it nhat mot nghiem thuoc V 2j = CO dung nghiem thuoc 71 2V2+2 -1 = n yl2 , • cos—>cos—= — va 4'2 / ~ \ Ax^{7i;2n) ^ ' • sin 4x < nen ta dugc 2cos^>V2>2014sin4xo + V Xet phuomg trinh sau log, [3 + '^-'^^^ = log^ (3"^ +1) 263 Dat log,(3 + = log,(3" + l) = t thi 3+Vy+T = 5' 3^+1 = 4' Suyra + 2' = ' « — + — = l5, l5J Ve trai la ham nghich bi^n nen phuong trinh tren c6 khong qua nghiem Thu true tiep, ta t h i y / = thoa man K h i do, jc = Vay phuomg trinh da cho c6 nghiem nhit la x = \ Xet phuomg trinh sau 3'°^''' = jc^ - DiSu kien x>0 R6 rang phuomg trinh chi c6 nghiem vai x >1 - > <» x < - l Ta biSn d6i logj x = log, (x^ -1) = r Khi do, ta c6 V - l = 3' x^2' nen 4' = ' + l o — + — = UJ V6 trai la ham nghich biSn nen phuomg trinh tren c6 khong qua nghiem Thu true tiep, ta thay / = thoa man K h i do, x = Vay phuomg trinh da cho c6 nghiem nhat la jc = Phuomg phap danh gia Nhan xet: Khi viec bien doi dai so true tiep khong tdc dung nita thi ta c6 the dung cdch ddnh gid cdc ve cua phuomg trinh thong qua nhitng dai luang trung gian Cdch tuomg doi nhe nhdng horn, chua cdn doi hoi viec ddnh gid ham so vd thdm chi nhieu hieu qud hem cdc dung ham so Vi du Giai phuomg trinh log2(2x^+4x + 2)-log2 x = l + x - x ^ L&igidL *) Dieu kien x > Ta bien doi nhu sau log^ 2x + - + = - ( l - x ) ^ X Dethay 2x + - + > - + = nen log^ 2x + - + > X X Dang thuc xay x = Dong thai, ve phai khong vugt qua nen dang thuc phai xay Vay phuomg trinh da cho c6 nghiem nhat la x = 264 Vi du Giai b i t phuong trinh logj JC + logj (jc +1) > log^ (x + 2) + log, (x + 3) L&igiaL *) DiSu kien x > De thay neu x = thi hai ve cua b i t phuong trinh bSng Ta se xet them cac truong hop sau: X x+2 x+1 x+3 - Neu X > thi — > > 1, > > 1, ta c6 day bat dang thuc sau X x+2 x+2 logj - > log2 ^ - > log4 ^ - ^ log2 X > log^ (x + ) logj > logj > log; =^ logjCx +1) > logjCx + 3) Cong hai bat dang thuc lai, ta dugc logj X + logj (x +1) > log4 (x + 2) + log, (x + 3), thoa man - Neu X < thi bat dang thuc d6i chiSu nen khong thoa man Vay bat phuong trinh da cho c6 nghiem la x e [2;+c») Vi du Giai phuong trinh sau x+2 LMgiai Ta c6 danh gia logj(2x^ + 42) > log^ ^2 = Ta se chung minh rang ^^^<i«x^+2>2^/77T«f^/77l-lf >0 x^+2 V / Do ding thuc phai xay ra, tuc la x = Vay X = la nghiem nhit cua phuong trinh da cho Vi du Giai phuong trinh sau logj (x^ + ^) ~ log2 x = 3x^ - x \ LoigidL *) Dieu kien: x > Phuong trinh da cho tuong duong: log x^+1 = 3x'-2x\ x^+1 Ap dung bat dang thuc Cauchy, ta c6: logj X 2x > log2 — = logj = X Xet ham s6 / ( x ) = 3x^ - 2x^ tren (0; +oo) Taco: / ' ( x ) = 6x - x ' = 6x(l - x) x>0 «>x = l 6x(l-x) = 265 Ta cung c6: /(1) = 1, lim /(x) = 0, lim /(x) = -oo nen x = \ cue dai eiia ham s6f{x) Suy / ( x ) < /(I) = Tu ta eo: log^ > - 2x\ X Dang thuexayra x = Vay phuong trinh da cho c6 nghiem vhit x = \ Vi du Giai phuong trinh log,_^ (2x) + log^ (2 - 2x) = L&igidL *)Bieukien 0<x<\ Xet phuong trinh log,_^ (2x) + log^ (2 - 2x) = Dat o = log2 (1 - x), Z> = log2 X Ta CO a + b = logjCl - x ) + log2 X = log2 [ x ( l - x ) ] < log2 = Phuong trinh da cho tuong duong v6i ' -2 10g2 + 10g2 X ^ l0g2 + l0g2 (1 - ^ ) ^ Q log2(l-x) log2X \ b l + a „ 2 b a ^{a + \f +ib + \f -{a + b + 2) = Do a + A + < nen (a + lf +(J) + \f -{a + b + 2)>0 va dang thue ciing xay va chi a = b = -1 hay logjO -x) = log2 x = -1 <^ x = ^ Thu lai, ta thay thoa man Vay phuong trinh da cho c6 nghiem nhat la = ^ • Vi du Hoi eo v6 han hay hiiu han so thue x thoa man dang thuc • ^ l o g X = sin(5;rx)? Lo'/^/fli *) Dieu kien xac dinh X > Dat 5;rx = >'>0 thi - l o g — = s i n ; ^ o log2 =sin>; STT {STT) Ta thay ve phai la mot ham so c6 thi hinh sin, vl trai la mot duong cong tiem can voi true tung va y tang du lom, gia su den gia tri y^ thi gia tri ma ham so nhan dugc se Ion hon 1, tuc la hai thi ciia ham so hai ve se khong diem nao chung nua 266 Hem nua, tren mien (O;^^), d6 thi ham s6 log^ y se cat cac duong hinh sin tai mot so hiru han diem va vi vay, phucmg trinh da cho c6 hiiu han nghiem Vi du Cho so thirc a > Hay tim t i t ca cac bo ba s6 thirc {x, y, z) vai |>'| > thoaman log^ {xy) + log, {x'Z + xyzf + 8+ ^4z-/ = Lfri giai *) Dieu Icien xac dinh xy > 0,4z-y^ > 0,x^y^ +xyz ^ <» xy > 0,x^y^ +zi^0,4z>y^, nen z > - va dieu kien tren c6 the viet lai la y nhimg |>;| > >l,z>—,xy>0 log^(xy) + log„ (x'Z +xyzy+ i l J ^ Z Z l > log^ (xy) + log, (x'y' + ^ ) + = = log^ (x;;) + log, (xy) + log, (x'y' + 7) + > log^ (xy) + log, (xy) + log, (xy) + = (log,(xy) + 2)^>0 ; (do a > nen log,(xV^ loga(2^^^>'^ ~ ) = log,(xy) nhu bien d6i a tren) Dang thuc phai xay nen ta c6 he sau z - / =0 xy = log„(^;^) + = 4z = / =1 xy = 1 Z - — /=1 1 xy = - = a' / a^yfl 1., , vol ' ' '•4J V thi khong c6 bo nao thoa man de bai Bai tap phan Bai Giai phuong trinh logj [yJx-2 + 4J = logj +8 Bai Giai phuong trinh 3'°^^'''"'^ = 2'°^'^''^" +1 Bai Giai phuong trinh hi(x^ + x +1) + x'* + x = 267 Bai Tim a de phuong trinh sau c6 dung nghiem: I log, (x^ + 4x + 6) 4- [S]'^'"' log, 22-|;t-sino+l| Bai Giai phuong trinh ^ log^ ^ —2(^x-sma ^= yjl" / V+\ log^^j / + l +1] =0 2-yJX Bai Giai bat phuong trinh logjCl + 2"^) 3^> +log = V^ + V ^ Bai Giai phuong trinh logj (x^ - 2x + 3^ + log x+—+\ Hirong dan giai bai tap phan Xet phuong trinh log^ ( V x - + 4j = log V + Dieu kien jc > Ta th4y log^ ( V x - + 4) > log^ = + <log39 = Ngoai ra, logj + 72^1 Do do, dang thuc phai xay o ca hai danh gia nen JC = Vay phuong trinh da cho c6 nghiem nhat la x = D i k kien 3' - > <=> x > Dat a = logj 3,y = 2' Phuong trinh da cho tuong duomg voi (3^ - ^ +1)'°^'^+\c>iy''-iy ={y + iy+\^ ((y" -1)" -1)" - = Xet ham s6 f(t) = t''-\,t>0 thi phuong trinh tren chinh la f{f{f{y))) = y De dang thay rang / ( dong biSn va /(2) = nen /(0>r,V/>2,/(0</,0</<2 Suy phuong trinh /(/(/(y))) =^ y c6 nghiem nhit la >' = 2, suy x = Vay phuong trinh da cho c6 nghiem la x = Xet phuong trinh In(x^ + x +1) + x'* + x = Ta xet cac truong hop: - N6u x(x +1) > thi ta c6: In(x'+x + l) + x ' + x = ln[x(x + l) + l] + x(x + l ) ( x ' - x + l)>lnl = "x = Dang thiic xay x(x +1) = x = - r Do phuong trinh c6 hai nghiem x = 0, x = - 268 - N6u x(x +1) < 0, ap dung hk ding thuc e' > / + » ln(/ +1) < /, > - , ta c6: l n ( x ^ + x + l ) + x'' + x = l n [ x ( x + l) + l ] + jc(x + l ) ( x ^ -x + \) <x(x+\) + xix + l)[x^-x + \)<0 Do phuomg trinh v6 nghiem Tir cac truong hop tren, ta c6 nghiem cua phuong trinh la jc = 0, x = - Dat x^ +4x + = M , ( | x - s i n a + l| + l ) = V , taco =3 Phuong trinh da cho tuomg duoTig vod 3-3-3-" log.v Mog^« + log^ - = « Mog^M = Mog^v<=>3 = V log^ u Ta thay rang neu M > v thi ^ > <1 nen khong thoa man log," Tuomg tu neu u<v Do do, ta phai c6 M = v hay x^ + 4x + = (|x - sin a +1| +1) <=> x^ + 4x + = |x - sin a + | Dat x + = y t h i t a c o =2|_y-sina-l| Xet truang hop: -Neu >'-sina-l<0<=>>^<sina + l thi = -2{y - sin a -1) + 2>' - - sin a = Ta CO A' = l + (2 + 2sina) = + 2sina>0 nen phuomg trinh luon c6 nghiem phan biet la = - ± V3 + 2sina Chii y r i n g - + V3 + 2sina < sin a + « V3 + 2sina < sin a + o + sin a < sin^ a + 4sin a + «> sin^ a + 2sina + l>0<=> (sin a +1)^ > Do do, truang hop nay, phuomg trinh c6 hai nghiem thoa man dieu kien -Neu j ; - s i n a - l > < = > > ' > s i n a + l thi / =2(;;-sina-l)<» - ^ +2 + 2sina = (*) Ta can tim sin a cho phuomg trinh c6 dung mot nghiem thoa J > sin a +1 Taco A' = l - ( + 2sina) = - l - s i n a > < = > s i n a < - - 269 Ta xet cac truong hap: - Neu sin a = thi ta dugc - +1 = <=> >' = < > ( i • s,)-! i<§M - , - Neu - < sina < - - thi phuong trinh c6 nghiem phan biet CO 3^, < sin a +1 < ^ < ^ < y^ va ta can {y\ sin o - \){y^ - sin a -1) < Chu y r i n g phuong trinh (*) cung c6 dang {y-y^){y-y2) = ^ nen diSu kien tren tuong duong (sin a + \f - 2(sin a + l) + + 2sino<0<::> sin^ a + sin a +1 < , v6 ly -n a = — + k27i: Tir suy gia tri thoa man de bai la sin a = - — <=> a =^ + )t2;r Dieu kien < jc < Ta c6 • »log^^,(2^+l)''=log^^,(2-V^)"'<»2^+l = - V ^ « ^ + V ^ = l Do x>0 nen 2"^ >2'' = l , V x >0 nen 2'^+Vx >1 vadSngthucxay rakhi x = Thu lai ta thay thoa man Vay phuong trinh da cho c6 nghiem nhdt la JC = ' ^ - ' I ^ * Ta se chung minh log2(l + 2'')>logj 3'+(V2) voi moi xeR Ta xet truong hop sau: - Neu x>Q thi ta c6 " ••' ^72^ log2(l + 2^) > log3(3^ +(72)') <=> log^ 2^(1 + ^ ) >10g3 3^(1 + ) v ' ^ o x + log^ 1+>X 2^ J Tathdyrang log^ + + l0g3 »10g2 V 1^ ' >l0g3 ^ r f \ 1+- >log3 1+ V ^ / 1+- ^ >log3 + V ^J V r V2 , nen b i t dSng thuc , t ^ i ^ / cuoi tren dung va bat dang thuc da cho dung voi x > , -Ngu ;c<0 thi 2'>3M>(>^)'=>2^+1>3^+(V2)'>0 Dodo log^ (2^+1) >log2 ^ + ( V H > I o g y+[yf2j Bat dang thuc can chung minh cung dung Vay hit phuong trinh da cho c6 nghiem la x e Xet phuong trinh logj (x^ - 2x + 3) + log, x + — + = V x + V - x X Dieukien < x < Taco X ' - J C + = ( X - ) ' + > , X + - + 1>2 + = nen , l o g ( x ' - x + 3) + log3 x + —+ >log2 + log33 = X Ngoaira,tacungc6 (Vx + V - x ) =2 + ^ x ( - x ) =2 + V l - ( l - x ) ^ <2 + = nen Vx + V - x < Tu suy dSng thuc phai xay va x = la nghiem nhat cua phuong trinh Phuong phap ham so don di^u Nhan xet: Doi vai cdc phuong trinh, bat phucmg trinh sieu viet, chua ddng than cd da thuc, ham so mU vd hdm so logarit, viec dung tinh dan dieu cua ham so gdn nhu Id dieu tdt yeu Trong mot so truang hap, hdm ban ddu chua hdn da dong bien, nghich bien tren todn mien xdc dinh vd ta cdn ket hap them mot so tinh chat nita de thu hep mien xdc dinh Igi giiip cho hdm s6 cdn xet thuc syt dan dieu vd suy so nghiem cua phuang trinh, bdt phuang trinh Vi du 1, Giai phuong trinh x^ + 2x + + ln(x^ + x +1) = , o p'l L&igidL Xet ham s6 / ( x ) = x^ + 2x + + ln(x^ + x +1), x e M Ta c6 f\x) = 3x'+2 + X 2x + l + X + ( x ' + ) ( x ' + x + l) + (2x + l) X + X +1 ( x ' + x ' + x ' ) + ( x ' + x + 3) X^ + X + Suy ham so dong bien va phuong trinh f(x) ; • >0,Vx( = c6 khong qua mot nghiem Honnua / ( - I ) = ( - ) ' + ( - l ) + + l n ( ( - l ) ' + ( - l ) + l ) = nen phuong trinh / ( x ) = CO nghiem nhdt la x = - Vay phuong trinh da cho c6 nghiem la x = - ,, ^ 271 V i du Giai phucmg trinh •x'"^^" + log 2) Ldi giai *) DiSu kien < jc Dat r = log^ x ^ x = 2' Thay vac phuong trinh, ta c6 ( ' ) ' + ( / - ) ' = 2" « 2''"' + +1 = 2" + 2t Xet ham s6 f{y) = 2^ + >',>^ e M Ta thay f\y) = 2^.1n +1 > nen ham d6ng b i l n Dang thuc cuoi a tren chinh la / ( / ' + l ) = / ( nen +1 = 2/<=> ( / - I ) ' = « / = l Tir suy x = la nghiem cua phucmg trinh da cho V i du Tim nghiem duomg cua phuong trinh xln + LMgidL + X = xMn + +"1 *) Phuong trinh da cho viet lai xln - l = x x' In + V X -1 J X < » x xln V V -1 = x xj 1.1 xMn + x \ -1 A Xet ham s6 / ( / ) = t /In -1 vai / > -2 Taco / ' ( / ) = (2/+ 1) In Ta se chung minh rang f'{t) > hay In Xet ham so g{\) = In 2t + l tJ Ta CO e'(t) = — z — + 2/ + vai moi / > ,t>0 = —; -1 '- T- < nen g(t) la ham giam tren (0; - H » ) Mat khac, ta cung eo lim g{t) = lim In + •— = Suy g(t) > hay / ' ( / ) > 0, tuc la / ( t ) la ham s6 dong bien 272 Phuong trinh ban dhu chinh la /(jc) = / ( x ^ ) <:> x = <=> x = 1, x > ^ Vay phuong trinh da cho c6 nghiem nhit la x = V i du Giai phuong trinh 'T'' = + log, (6x - ) \ L&igidL *) Dieukienxac dinh x^ — Dat x - l = a=>a5^-— Ta CO phuong trinh 7" = + log, (6a +1) TatiSptucdat log,(6a +!) = /=> a + = 7' ¥ 7° =1 + 6/ Ta duoc he sau \ 7" =6a-6t [7" =6^ + <^T +6t = 7° +6a Xet ham s6 f{y) = 7^ + 6>^, >^ e M Ta c6 f'(y) = n n + > nen day la ham dong bien Dang thuc cuoi tren chinh la / ( / ) = / ( a ) nen ta c6 duoc a = t hay ^ = + ' • ' ' Day la phuong trinh Bernoulli xet a tren nen no c6 nghiem la a = v a = 1, tuong ung, ta tim duoc hai nghiem cua phuong trinh ban dhx la x = v x = Thu lai thay thoa Vay phuong trinh da cho c6 hai nghiem la x = 1, x = V i du Giai phuong trinh sau log3(2x +1) + log5(4x +1) + log7(6x +1) = 3x L&igidL *) Dieu kien xac dinh: x > Xet ham s6 / ( x ) = log3(2x + l) + log5(4x + l) + log7(6x + l ) - x lientuctren ;+oo Taco: f\x) = f"(x) = -+- (2x + l)ln3 -+ • (4x + l)ln5 16 z (2x + l)Mn3 (6x + l)ln7 (4x + l)Mn5 -3 36 < voi moi x > — (6x + l)Mn36 ' Matkhac: /'(O) = ^ + ^ ^ + — - > 0, / ' ( l ) = - l - + - i - + -A In3 ln5 ln7 31n3 51n5 71n7 <0 Do phuong trinh /'(x) = c6 nghiem nhit tren Suy phuong trinh / ( x ) = c6 khong qua hai nghiem tren Ta cung C O / ( O ) = / ( I ) = Vay phuong trinh da cho c6 hai nghiem la x = 0; x = 273 Vi du Giai phvromg trinh sau log, ( ' + 2) = logj (6' +19) L&i gidL *) Xet phuong trinh log, (V' + 2) = log; (6" +19) Neu JC < thi ta c6: logj (7" + 2) < log, = < log, 19; log, 19 > log, (6" +19), phuong trinh v6 nghiem Do do, ta chi can xet jc > la du Xet ham so / ( x ) = log3(7''+2)-log5(6''+19) lien tuc tren [0;+oo), taco: fix) = (6"+19)ln5' (7^+2)ln3 6Mn6 7Mn7 Ta cung CO I n > l n > , I n > l n > = > — > — > In3 ln5 r ln6 19-7^-2-6^ 6' > voi moi x > Suyra f'(x)>^^^ hi5 In5 (7^+2)(6^+19) 6'+19 d6ng biSn tren [0;+oo) ma /(1) = nen phuong trinh da c6 r r+2 Suy f{x) nghiem x-l Vay phuong trinh da cho c6 nghiem nhat x-l V i du Giai b i t phuong trinh sau xlog2 JC > ( x - l ) ^ L&i gidL *) DiSu kien x > Do (jc-1)^ > nen ta CO jclog2 X > => log2 JC > JC > Xet ham s6 / ( x ) = xlog^ x - ( x - l ) ^ x > Ta c6 / ' ( x ) = log2X +In- i2- - ( x - l ) , r(x) = X - ^ - < - ^ - < In In Do do, phucmg trinh f{x) = c6 khong qua nghiem Ta cung thSy ring / ( I ) = / ( ) = nen phucmg trinh f{x) = c6 dung nghiem x = 1, x = D e t h i y t r e n c a c m i ^ n [l;2],(2;+co) t h i d d u c u a / ( x ) khongddi Thu true t i l p , ta thdy / ( x ) > 0,x e [1;2] va / ( x ) < voi xe(2;+oo) Vay nghiem cua bat phuong trinh da cho la x e [1; 2] ^ • Bai tap phan (x —1)^ Bai Giai phuong trinh 81og,^^ ^ = x^-18x-31 2x + l Bai Giai phuong trinh log2 (2" + 4) + log3 (4^^' +17) = 274 I- Bai Tim so thuc m dk phuong trinh sau c6 nghiem thuc doan (/«-l)log^(x-2)^+4(/«-5)log Bai Giai phuong trinh 4(x - 2)[log2 (x • + 4m-4 =0 - 3) + log, (x - 2)] = 15(x +1) Bai Giai hit phuong trinh log, | l + 2yJx^-x + 2^ + log, (x^ - x + ? ) < Bai Giai bdt phuong trinh + 3''-2x-l Bai Giai phuong trinh Vx + SyJT^ >Q = log, ((3 - x ) ' (2x +1)) ) t ^ ' , ^ tf a H u a n g dan giai bai tap ph'an Xet phuong trinh log 2x + l = x^ -18x - 31 Dieu kien xac dinh: • x> — x^l Ta c6: - logjCx - ) ' - Iog2 (2x +1)] = (x - ) ' - 8(2x +1) - 24 « (x - ) ' + log^ (x - ) ' = 8(2x +1) + 24 + log^ (2x +1) « (x - ) ' + logj (x -1)^ = 8(2x +1) + log2 8(2x +1) Xet ham so / ( O = / + logj / tren (0; +oo) Ta c6: f'(t) = 1+ /in > voi moi />0 Suy / ( d6ng biSn tren (0; +oo) Phuong trinh ban dku chinh la: f[ix-\y] = f [8(2x + ) ] « ( x - ) ' = 8(2x +1) <=>x - x - = « "x = - V 2 _x = + 2>/22 ' : - - Ta thay rang cac nghiem tim dugc deu thoa man diSu kien ban dku Vay phuong trinh da cho c6 hai nghiem la x = + 2V22,x = - 2V22 [2^+4 = 2" Xet phuong trinh log^ [2" + 4) + log^ (4'^' +17) = Dat , taco: [4^"'+17 = 3' Tu thay vao phucmg trinh da cho ta dugc: 4(2" - 4)^ - 3^'° +17 = Xet ham s6 / ( a ) = ( " - ) ' - ' " " + tren M, taco: / ' ( a ) = ( " - ) - " - l n + 3'-"-ln3>0, V a e M Suy f{a) d6ng bifin tren M Mat khac, / ( ) = nen f{a) = c6 nghiem nhat a = » " = ' - = « x = Vay phuang trinh da cho c6 nghiem nhat x = Phuang trinh da cho tuang duang vai: ( w - l ) l o g ^ ( x - ) - ( / « - ) l o g , ( x - ) + / M - l = Dat / = l o g , ( x - ) , x € te[-l;l Ta thu duac phuang trinh m = f(t) = Ta CO fit) = 4/^-4 ( / ^ - / + 1) /'-5/ + -;/'(0 = < » / = ±l Ta CO /(1) = -3, / ( - I ) = — Lap bang bien thien cua /(0 tren doan [ - ; l ] , ta thSy / ( r ) lien tuc va nghich bien tren doan [ - ; nen m e -4 thoa man dh bai Xet phuang trinh 4(jc - 2)[log2 (x - 3) + logj(x - 2)] = 15(x +1) Dieukien x > Phuang trinh da cho c6 the viet lai la log2(x-3) + log3(x-2) = 15 x + \ x-2 Ta xet dao ham cua cac ham so tuang ung a m6i ve cua phuang trinh (log, (X - 3) + log3(x - 2))' = I (x-2y x-2 -3 x +1 I + (x-3)ln2 va > (x-2)ln3 < vai x > Suy ve trai la ham so dong bien theo x , ve phai la ham nghich bien theo x nen phuang trinh da cho c6 khong qua mot nghiem Thay x = 11 vao phuang trinh thi thay thoa man Vay phuang trinh da cho c6 nghiem nhat la x = 11 276 Xet bit phuomg trinh log, | l + 2ylx^ - x + 2J + log, (x^ - x + ? ) < Dat t = yjx^-x + 2>0,taCO x^-x + = t^+5 Do bit phuong trinh da cho tra logj (1 + 2t) + log, (t^+5)<2 Xet ham s6 / ( = log, (1 + 2t) + log, (t^+5)-2 vai / > 2/ ' ' +— > thi day la ham dong bien (l + 201n5 ( / ' + ) l n ^ Ngoai ra, ta cung c6 / ( ) = nen / ( r ) < vai moi t<2 Ta CO f'(t) = Khi d6,tac6 VJC^^^JC + < < = > - x - < - < x < Vay nghiem cua bat phuomg trinh da cho la x e (-1;2) , , , ln(5 + x ) - l n ( - x ) ^ -^-^ i f3'-2x-1^0 Xet bat phuomg trinh — > Dieu kien < ^ ^ y-2x-\ [ - < x < Ta thay phuomg trinh 3"" - x - l = c6 dang Bernoulli nen c6 dung nghiem la va 1, ngoai ta cung c6 ' ' - x - l < tren (0;1) va ' ' - x - l > tren (-c«;0), (!;+«)) v'.u' u Ta CO ln(5 + x) - ln(5 - x) = « l n ( + x) = ln(5 -x)<=i>5 + x = - x o x = Ngoai ra, ta ciing c6 ln(5 + x ) - l n ( - x ) > vai x > va ln(5 + x ) - l n ( - x ) < vai x < Tu cac dieu tren, ta thay nghiem ciia bat phuomg trinh da cho la x e (1;5) Xet phuomg trinh Vx + sVT^ = log, ((3 - x / (2x +1) Di^u kien < x < Truac hk, ta se chung minh rSng vai moi / e [0;1] thi />log3 (2/^+1) That vay, xet ham s6 / ( O = 3' - 2/^ - , / e [0;1] thi /'(O = 3'In - / , / • ( / ) = 3'(In ) ' - Do / < nen /"(t) < 3(ln 3)^ - < va phuomg trinh / ( / ) - c6 khong qua nghiem Ta cung c6 /(O) = /(1) = nen phuomg trinh / ( / ) = c6 dung nghiem Xet d i u cua hiku thiic / ( / ) tren [0;1], ta c6 3' > 2/^ + « / > logj (2/^ +1) vai moi /G[0;1] T u d o s u y r a Vx + 3Vr^>log3(2x + l) + 31og3(3-2x) = log3((3-2x)'(2x + l ) 277 Dang thuc phai xay ra, tuc la x = 0, x = Vay phuong trinh da cho c6 nghiem la x = 0, x = _ V i du tong hgfp s r * \ Trong phdn nay, ta se xem xet mot so bai tap tong hap ve phuong trinh, bat phuong trinh, he phuong trinh mil lien quan den cdc dang Todn da neu Vi du Giai phuong trinh sau +A" Xog^ x = 0, _ L&i giai *) Dieu kien A: > Ta thay neu x > thi logj x > va 4"^ > nen ve trai duong, khong thoa man Ta chi cin xet < X < Xet ham so / ( x ) = + 4Mog2 x, x e (0; 1) thi / ' ( x ) = 4"ln41og2X + - 4- xln2 Ta can chiing minh / ' ( x ) > 0, Vx e (0; 1) 4" r n ln2 ln4-lnx + — X In • In x + - > 0, Vx e (0; 1) ' ' ^ V Dat — = >'>! thi dua bat dang thuc tren ve ln4-ln —+ >;>0<=>-^>ln4 X y \ny = 0<=>>' = ^ nen khao sat Dat g{y) = - ^ vai ;;>1 thi g\y)=^^—-,g'(>') ln>' ln>' ham so tren mien (l;+oo), ta c6 g{y) > g(e) = e D l thiy e > nen > 2^ = => e > ln4 Do > ln4,V>' > ln_y Dodo / ' ( x ) > 0,Vx€ (0;1) nen ham so / ( x ) da neu dong bien tren (0;l) Tacungco / = nen phuong trinh / ( x ) = c6 nghiem nhat la x = - Vi du Tim s6 nghiem cua phuong trinh sau theo tham so k: (l-x)ln = 2kx-2k + \ L&igidL Ta xet cac truong hop sau: - Neu k = thi thay vao phuong trinh, ta dugc (1 - x) In - = 0, v6 nghiem - Vdri ^ 9t 0, dat - x) = / thi ta dugc phuong trinh / In e Xet ham s6 f{t) = t I n p y , ^ ^ -'- kien-1 < / <1 = 2k vai diSu •' (-1;1) thi d l thiy /(O) = va lim/(t) = +oo 278 Taco / ' ( O = l n j ^ + : j ^ , / ' ( ) = va f''(t) = j^-^>0 n e n / = Olacirc | tiku cua ham s6 da cho va / ( / ) nghich biSn khoang ( - ; 0) va dong b i l n tren khoang lai Do do, k<0, phuong trinh da cho v6 nghiem va A: > thi phuong trinh da cho c6 hai nghiem phan biet Vi du Giai cac bat phuong trinh sau a) log, (logj(9^ - 72)) < (Di DH khSi B 2002) b) (log, + log^ x') log2 V2^ > (DiDH LaigiaL khoi A 2007) a) Xet hk phuong trinh log, (log3(9' - 72)) < ro<x^i Dieu kien \ • [log3(9^-72)>0 0<X9tl <=> < [9^-72>l < » X > lOgg 73 Ta CO log3(9^ - 72) < X <:> 9"^ - 72 < \ D a t / = 3^>0 nen ^ ' - / - < < » ( / + ) ( / - ) < « / < Suy 3"^ < « X < Do nghiem cua bat phuong trinh da cho la log, 73 < x < b) Xet bat phuong trinh (log, + log4 Bdt phuong trinh tuong duong vod + log2X ^ l o g j x > < ^ loggX ) logj ^llx > Dieu kien < x 9^ l + log2X (log2X + l)>0 log2X Dat / = logj X ?t thi - + t (/ + ) > « Neu / > thi logj x > ^3 + /^^ r/(/+i)>o (/ + l ) > « - ^ />0 /<-l x > Ngu t<-\i logj X < - « X < - Vay nghiem cua bat phuong trinh da cho la x e Vi du Cho a,b>\ u(l;+co) Chung minh rSng phuong trinh log' y + log^ X + log^ X log^ CO nghiem va dat la (x,, y^), (xj, - log, y log, x - ( l o g ^ + log^ x) + 80 = ) thi x, + Xj + + ^2 > (Tuyen tap 45 nam tap chi THTT) 279 L&i gidL *)Bihukienxac d\r)h x,y>0 Dat u = log„ y,v = log^ x ta dugc u'+v'+ 3M'V' - 8MV - 6(M^ + M'+V' MV = =8 (W-V)' ) + 80 = (M' + V ' - 8)^ + (MV - 4)' = =0 wv = Giai he nay, ta thu dugc w = v = ±2, tir ta dugc x^ +y^+X2+y2 =a +b + —+ —>4 a Vi du Giai bat phuorng trinh log, yl2x^-3x + \, (x +1) • L&i gidL *) Dieu kien xac dinh 0<x + l5^1 <=>x€(-l;0)u TathdyrSng log, \/2x^-3x + l > - x log, (x +1) > X +1 + l < » 0< j c < - va <1 fo-il <=> X < 3) , ta thay bit phuong trinh '2J hiSn nhien dung Vai X e (-1;0) thi bdt phuang trinh sai Vai x€ thi bat phuang trinh da cho tuang duang vai V2x^-3x + l > X + 1>0<:>- x>-l -l<x<0 x>5 x'-5x>0 K6t hgp vai truong hgp dang xet, ta c6 x > 3^ xe 0;- u 1• — u(5;+c») l 2j V 2) Vi du Tim ik ca cac bg s6 thuc (x,>') thoa man d6ng thai ^P-2H->og,5 4^ _ ^^-(,,4) +(^4-3)^ <8 Ldigiau * ) T a c r - j c - > nen 5-(y-4) > 3-iog,5 ^ 5-iog,3 = 5-' ^ + 4) > _ i ^ ^ < _3 Dodo H - | > ' - +{y + 2f =-Ay + {y-\) + {y + 3,f =-Zy-\ {y + 7>f > - l + = =0 DSng thuc phai xay ra, turc la x =-l,x=3 <=> y = -7> y^-3 Vay CO tat ca bo thoa man la {x;y) = (-l;-3),(3;-3) Vi du Giai phuong trinh log^ (11-3^) = log, L&igidL *) Dieu kien {1-2') 11-3^ > 7-2'>0 • Dat y = log3(7-2")r:>2" +3^ = ^ < nen; log,(ll-3^)<2=>3^>7^x>- Xet ham so f(x) = l o g ^ O l - ^ ) - l o g ( - " ) , - < x < Ta c6 f'ix) = Do X > —: -3Mn3 2Mn2 -3" T —— + ^ < + ( l l - ^ ) l n ( - ^ ) l n 11-3^ 7-2^ ll-2'-7-3' (ll-3^)(7-2^) 11 ^3 ^ > —^11.2^-7.3^ < , s u y r a / ' ( x ) < , V x e - ; v2y V2 Ham so nghich bien nen c6 khong qua mot nghiem Dong thoi / ( ) = nen x = la nghiem nh4t cua phuong trinh / ( x ) = Vay phuong trinh da cho c6 nghiem nhit la x = Bai tap tong hgp Chuang Bai Giai phuong trinh sau (x +1) log 4' = xlog(x + 2'*^) > Bai a) Chung minh rang moi phuong trinh sau day c6 nghiem nhat CQSX = X (1) sin(cos x) = X (2) cos(sinx) = x (3) b) Goi a,p,/ ' Ian lugt la nghiem cua cac phuong trinh (1), (2) ''3) Chung minh rang/ia f t < y^;''In or < c!r>9 In 281 Bai Giai bat phuomg trinh sau ^ ^ " § ^ jc-l ^— ^Q 2x-\ Bai T i m tat ca gia t r i a de phuomg trinh log^ilS'' -log^ a) = x c6 nghiem nhat Bai Tim m de phuong trinh 4(log2 Vx)^ - log, x + w = c6 nghiem thuoc (0; 1) Bai Chung minh rang cac bo ix,y,z) thoa man J'"'"'"-^"'"^ [xyz=30 ^^,c6it nh4t bo thoa man dSng thuc logj jclogj jylcg, z = Bai Giai phuomg trinh sau log2x^.4 + log3^,^, + log^^,^^, = , ^ • + ^log2 (x' + 2) log3 (x' + 3) log, (x' + 5) Bai Giai phuomg trinh sau ^-s'^^'^^J _^iog,i5 ^ Huang dan giai bai tap tong hgfp Chuong Bai Phuomg trinh (x +1) log 4"^ = x log(x + 2"^^) tuong duong voi xlog4^"'=xlog(x + ^ ^ ' ) « x = v ^ " ' = x + 2^"' Ta xet phuomg trinh 4'^' = ' ^ ' + x , d a t x + l = :>;,tac6 ^ - ^ - > ' + ! = Xethams6 / ( ; ^ ) = 4^ - ^ + l Ta c6 f'{y) = 4nn4-2'\n2-\ De thay rang phuomg trinh f'(y) = c6 dung mot nghiem k va f'(y) duong trenmi^n ik,+oo) vaamtrenmiSn {-00; k) / - , \3 = n n - M n - l = ln2(22 - ^ - , Ta Cling thay rang / ' — > / ' ,4 /1 > 10 m a l n > l o g > — va ^ - ^ > — - = — h a y / ' = ln2(22 - ^ ) - l > ^ 10 3 Do A: < — Ta ciing c6 /(>>) = ' - i + — y=^ f(y)>0,yy <— nen m>o Ro rang A: l a d i l m cue tieu cuaham s6 nen f(y)> f(k)>0 Suy phuomg trinh 4^ - 2^ - >' +1 = v6 nghiem Vay phuomg trinh da cho c6 nghiem nhat la x = 282 Bai a) Xet ham s6 tuong ung voi phuong trinh (1) la / ( x ) = x - cos x Do - < cosjc < nen ta chi can xet x e Taco / ' ( x ) = l + sinx>0,Vxe 2'2 2'2, nen day la ham dong bien Hon nua, / ( ) < , / ( l ) > va day la ham lien tuc nen phuong trinh / ( x ) = CO nghiem nh4t Ta c6 diSu phai chung minh Cac phuong trinh (2) va (3) dugc chung minh tuong tu b) De thay or, /?, 7^ G (0, l ) Bat dang thuc can chung minh tuong duong voi In y3 In Of In / ——< <—In/ Xet ham so g{i) = — , < / < , _ ^ Ta CO g'(0 - ^ ^1"^ > nen ham so dong bien tren (0,1) Do do, de chung minh b i t ding thuc cu6i tren, ta chung minh p <a <y Gia su P>a thi P = sin(cosP) <cosp <cosa = a, mau thuan; tuc la y5 <or t Gia sic /<a thl y-cos(sin7) Vay p <a <y hay —— < P a = cosy>cosa = a, mau thuan; tuc la a < / < —-<:> ya\np< ? Bai Xet b i t phuong trinh i ± i ^ x-1 Dieu kien xac dinh: x>0,x pyXna <ap\ny y ^^,x ^ < 2x-l .1 i^l 2x-4 thi / ( x ) , g ( x ) la cac ham s6 d6ng b i l n K i hieu / ( x ) = log3x; gix) = 2x-l tren m6i khoang thuoc tap xac dinh Ta xet cac truong hop sau: (1) Neu < X <— Khi bat phuong trinh tuong duong: / ( x ) > g ( x ) Ta c6: fix)<f < < < g(0) < g ( x ) i 283 Suy / ( x ) < g{x) voi moi x e V '2j Trong truong hop bat phuong trinh v6 nghiem (2) Neu ^ < jc < Khi bat phuong trinh tuong duong: /(jc) > Ta c6: = log3^>-0,5>-l>g(l)>g(x) Suy truong hop bat phuong trinh c6 nghiem la (3) Neu < ;c < 2.Khi bat phuong trinh tuong duong: /(;c) < g{x) Ta thiy / ( x ) > / ( l ) = = g(2)>gW Suy bat phuong trinh v6 nghiem (4) Neu < JC < Khi bat phuong trinh tuong duong: / ( x ) < g{x) Ta c6: f{x) > / ( ) = log3 > , > , = | = g(3) > g(x) Suy /(jc) > g(x) Do do, b i t phuong trinh v6 nghiem (5) Nea Khi bat phuong trinh tuong duong: / ( x ) < g(x) Ta c6: f(x) > / ( ) = > g(x) nen suy r a / ( x ) > gix) Do do, bat phuong trinh v6 nghiem T u cac truong hop tren, ta c6 nghiem cua b i t phuong trinh la ^ < x < Bai Phuong trinh da cho tuong duong voi 25^-log5a = 5^ <=> ' ^ - ^ - l o g , f l = « ''^^ [t'-t-\og,a =0 (*) Phuong trinh da cho c6 nghiem nhit (*) c6 dung nghiem duong hay -t = log; a CO dung nghiem duong Xet ham s6 f{t) = t^-t Taco: f\t) = 2t-\ voi t e [0; +oo) /'(0 = 0«/=i; /W = - ' / ( ) = 0- \2) Dua vao bang bien thien, ta suy phuong trinh / ( / ) = log5a c6 dung nghiem duong a>\ log5a>0 log.a^ a= Bai Dieu kien jc> Phuong trinh da cho tuomg duong vai log]X + logjx + m = 0; xe(0;\) Dat / = logj X Vi lim log^ x = -oo va lim log x = 0, nen xe(0;l)=>/e(-oo;0) « Ta c6: -t-m = 0, t <0<:> m = -t^ -t, t <0 Lap bang biSn thien cho ham s6 f{t) = -t -t vai t <0 de suy ket qua Vay gia tri m can tim la m<— Bai Xet he diSu kien r"^-^"*'^~^^.Dl thSy {x,y,z) = (2,3,5) va cac hoan xyz = 30 vi cua chung thoa man he phuong trinh da cho Chii y rang log, xlog, y log, z ^ ^"^^""^^"^ = vai moi bo hoan vi tren In21n31n5 Vay CO it nhat 3! = bo thoa man Bai Dat a = log^(x' +2),b = log,(x' + 3),c = l o g s + 5) thi a,b,c>\ Chu y rang log, , = ^— = ^-^ = —— +4 iog2(2x'+4) l + log2(x'+2) \ a Tuong ty log , = — - — \ ; log 5=— — \ ^"^^ + logj (x' + 3) + logs + 5) Ta se chung minh bdt ding thuc — ^ , + ^ - ^ > — - — vai a,b,c>\ •+ T> \ a<=> \ b ; l + c ^ l>+ abc + fl' l + b^~l + ab (l + a^)il + b^) l + ab thong qua bat dang (2 +thuc a'—+ ^6'sau)(1 + —+ ab) ^ >> ^2(1—+ a^+b^+ vai a,b>la^b^)(*) l + a' ++6'ab{a^l ++b^)>2 ab ^2 + a^+b^+2ab + 2a' +2b^ + 2aW That vay, <^{a-bf{ab-l)>0 Bit ding thuc cu6i dung1 nen (*) dugrc chung + a ' +minh 6' Do 1 1 ^ 2 ^ ^ - ; = > •l + abc l + a^• + + 6'r + l + c'r + l + abc > l + ; + l + c^yfab 285 nensuy r- + \ a' \ b' Tir suy log,, r- + l + c' r-> l + abc vm a,b,c>\ + W , + log,, , 5> + ^logj ix'+2) log3 (x' + 3) log3 {x' + 5) Dang thuc xay x = Vay phuong trinh da cho c6 nghiem la x = Bai Dieu kien < X Dat = a > 0, ta CO ' - - x'""^'' = x'"^'' • x = ox va Q Q Q Q Q\ Q\ Q ' a X = — Tir taCOphuorngtrinh x^-ax +—a^ = ( x - a ) ( x - a ) = • » 2a x= — Ta CO truong hop: - Neu ^ = ~ thi ta dugc X =• <=> logj X = logj • log3 X -1 o log3 X (1 - log3 5) = -1 108,3 <=> logj X = logj <=> X = ' jr - Neu X = ^ X= thi ta dugc 2x'°^'' ? r 2 « logj X = logj • logj X + log3 - » log3 x(l - logj 5) = log, - '°^'3 » logj X = l0g3 - <::> X = ' log, logj? _ , - Vay phuong trinh da cho c6 nghiem la x = ' , x = ' 286 QYaiHag P H l / d N G T R I N H , BAT PHlTCfNG T R I N H HtTU T I §1 Tam thufc, phi/cfng t r i n h , bfit phiTcfng t r i n h bac hai §2 Phtfcfng t r i n h , bat phifcfng t r i n h bSc cao phiicfng t r i n h , bat phifcfng t r i n h hOfU t i V i du tdng hgp Chifcfng 31 66 Chtftfng PHl/CfNG T R I N H , BAT P H l f d N G T R I N H V T I A Mot s6' dang phiicfng t r i n h co ban 80 B Phifcfng phdp giai 1) C^c phifcfng phdp dai so 2) Cdc phiicfng phdp liicmg gidc 82 gidi t i c h 3) V i du tdng hop Chi/cfng Biii tap tong hcrp Chiicfng 135 167 177 Chtfdng PHLfCfNG T R I N H , B A T P H l / d N G T R I N H MU A T6m tat ly thuyet 182 B Phirong phap giai 1) Phiicfng phap bieh ddi ve cung ccf s6' hoac lay logarit hai ve 184 2) Phtfcfng phap dat ^ n phu 190 3) Phiicfng phap phan tlch 197 4) Phiicfng phap danh gia 205 5) Phiicfng phap dung ham so 212 6) V i d u t n g hcfp 224 Bai tap tong hcfp Chiicfng 231 Chifrfng PHlTCft^G T R I N H , B A T PHtfdNG T R I N H L O G A R I T A T6m tat ly thuyg't 245 B Phifcfng phap giai 246 Phiicfng phap bien doi ve cung ccf s6' 246 Phiicfng phap dat an phu diTa ve dang dai so 251 Phi/cfng phap dat an phu di/a ve dang mu 258 PhiTcfng phap danh gia 264 Phiicfng phap ham so dcfn dieu 271 V i du tdng hcfp 278 287 y • y / / y y y / y y- y y ^ y • X \ \ \ Email: nhasachhongan@hotmail.com C N g u y e n Thi M i n h Khai - Q.1 - T P H C M DT: - 7 - 9 • F a x : • \ \ A (^^SK- ^amy ti^rri' e^bo: - 245 Tran N g u y e n Han - HP * DT: 3858699 - & P h a n B p i C h a u - H^i P h o n g *DT: - 04 Ly Thai To - TP Da - 259 Le Duan N i n g - TP Vinh *DT: - DT: - 15 Le T h a i T o - VTnh L o n g - DT: 3823421 ^ -< ISBN: 978-604-62-2269-9 \ 3554777 - 39-41 V Thi Sau - Can T h d * DT: • TTnh 1^ - T T C u C h i - T P H C M 3839599 3818891 *DT: \ 37924216 0907845219 \ 935092 763842 G i a : TO.OOOd