Chuyên đề bồi dưỡng học sinh giỏi giá trị lớn nhất, giá trị nhỏ nhất phan huy khải (phần 5)

Chuyen dg BDHSG Join g\i tr| Ifln nh^t va gia trj nh6 nhat - Phan Huy Khil (x;y)eD P = min< P; [(x; y)eD| (x; y)eD2 P > Cty TNHH MTV DWH Khang Vijt p i Tim gia tri I6n nhat va nho nha't cua ham so: (2) J ' + 4X^+3X^ f(x) = - Do iha'y D, - {(x; y ) : y = va x^ < } , va P = x^ (x; y) e D, vdi X e • (l + x ^ ) ' Tirdo suy ra; Hudiig dan giai max P = 3x = ± > / ; y = , •>>' (3) Goi m la gia trj y cua ham so: f(x) = (x;y)eD| P = o x = y==0.(4) u , + 4x^ H-Sx'' Khi phifdng trmh sau (an x): —- m l + 2x^ +x'* Xet (x; y) e D j Liic do: x^ + xy + y^ > 0, vi the: P= X -xy-3y / ^(x^+xy.y^) X + xy + y c6 nghipm Ta c6: Xet hai kha nang: Neu m = 3, (2) c6 dang x^ = 0, vay (2) c6 nghiem -xy-3y ^^:„2 2 x"^-xy-Sy"^ < x^ + x y + y Do m = la mot gi^ trj cua f(x) yj X , + - +1 s/ r +t+l t^-t-3 ce fa w "^"^ < m < y m-3 = > , d day S va P tiTdng tfng la long va tich hai nghiem m>— 2 0, 3.^:;^^;^0 Ta c6: (4) o - m >0 m-3 ww Tilfdosuyra: Dieu xay DoP = c Dc'n day gpi m la gia tri luy y cua ham so: f(t) = Neu m ^ 3, (2) c6 nghiem va chi phiTcfng trinh (m - 3)t^ + 2(m - 2)t + m - = ro X N2 /g / up t^-t-3 om " ' - " y - y , X,2• + xy + y (2) « ( m - 3)x* + 2(m - 2)x^ + m - = X Tac6: ' Ta P^3^ (1) (1) o + 4x^ + Sx" = m + 2mx^ + mx"* p ^ ^ x ^ j c j ^ J j ^ x - x y - y > X +xy+ y Do X + x y + y < , nen -, X G (iH-x^)' iL ie uO nT hi Da iH oc 01 / (x;y)eD| + 4x^+3x^ Khim = 3,thi(l)c6dang khix'-xy-3y' U+ V + w (1) gia tri ham so O/l Cty TNHH MTV DWH Khang Vi?t Chuyen BDHSG To^n gia tr| Ifln nhft v i g\i tr| nh6 nhflt - Phan Huy Khii (dal t = ( X + y + z)' Do < x + y + z < => < I < 1) 'I (x + y + z) + — + — + X v = k2W,k2 > '1-.' Ta c6: f'(t) = - — , va bang bien thien sau: y u, v, w la cac vectd cung phiTdng, cung chieu ' (3) :\iu>,ii int't ri^'^rhb ;:5iim) ,0,': 'mM' uw-i 1 = 81(x + y + z)^ + - ri n y zj x fi',; Theo bat dang thuTc Cosi, ta c6 VAb '1 ^1 iV —+ —+ - X z) y 1 h'un ipj'O - 1^ - >9 HUifiig dan giai ro Cho ham so f(x) = x + yfi-x^ trcn mien xac dinh cua no om c Hiiifng dan giai ok bo ce fa Tim gia tri nho nhat cua ham so f(x) = Vx^ - x + + Vx^ - V x + , vdi w e R ww X Hiidng dan giai f(x)= 1 Ap dung bat dang thuTc Cosi cd ban, la c : - + - + - > X y z x+y+z 81 (x + y + z) • ^ X 2J N2 (10) j Viet lai ham so' f(x) diTdi dang sau day j > (x + y + z)' + Tim gia trj Idn nhat va nho nhat cua ham so sach bien thien ham so nhi/ sau: ZJ mien gia trj ham so de giai bai toan trcn bai toan 1, §1, chU'dng I cuon Thay cho phiTdng phap bat dang thtfc ta c6 the sOr dung phiTdng phap chieU - „ Bai (De thi tuyen sink Dai hoc, Cao dang khoi B) phi/dng phap siir dung ba't dang thtfc de giai bai toan dat —+ — + X y , > Xem Idi giai ket hdp giiJa phifdng phap thi, hmh hoc va phifdng phap Trong bai trcn ta da kct hdp phiTdng phap suf dung vectd hinh hoc phang V i the (x + y + z) + i tr ; (8) (9) Nhirthe minP = >/82 o x = y = z = ^ + |y - 2| •' bai loan Xem Idi giai bai 13, §2, chi/dng cuon sach dong thcfi co dau bang (3), (5) (6) h': Ta kct hdp phi^dng phap suT dung vectd va chieu bicn ihien ham so' de giai (7) > 162 o x = y = z= - (l ^^.si^'K^'^Ai^-piih thi tuyen sinh Dai hoc, Caoddnfi khoi B) V; ZJ - Nh4n xet: ^ P = V ( X - l ) ^ +y^ + ^{X + lf+y^ /g i I'' Cho x y la cac so thifc tiiy y Tim gia tri nho nhat cua bicu thtfc (6) , TCr gia thiet < x + y + z < I = > 80(x + y + zf < 80 T i r ( ) ( ) ( ) ( ) t a c P > >/82 fiiii2.(De (5) TJ —+ — + X y Ta thu lai kct qua trcn up Ttr (5) (6) suy 81 ( X + y + /.Y + —+ —+ X y Thco bat dang thuTc Cosi cd ban thi (x + y + z) A Dau bang (9) xay 'i0M> (ii 81(x + y + /.)' + [ - + - + - >18(x + y + z) U y 7.) Vaytuf(lO) suyraminP= ^^min f(t) = - ( x + y + z)^ (4) — + — + - , Ta —+ — + s/ (x + y + /.)' + I'd) 1(1) Dc thay , t iL ie uO nT hi Da iH oc 01 / Dau bang (2) xay o X., ,1 t, ^ I (2) + X - + — Xet he true toa Oxy, va tren xet cac diem A (1) va C(x;0),xe K 81 Xet ham so f(l) = t^ + y, Khi tir (1), ta c6 f(x) = CA + CB d day < t < 245 Chuyen dg BDHSG ToAn gia tri I6n nhat va gia tri nh6 nhS't - Phan Huy Khai Cty TNHH MTV D W H Khang Vi§t R6 rang ta co C A + CB > A B , -> O M „ = OC + AB = V^^' siin45|^^N/2 I sin75" • +cos 30 ,0 73 V2 + 73 V Nhir the ta C O f(x) > N / ^ Vx G E Goi C„ = A B n Ox Ta c6 C„A + C B = 74-273 (2) AB =J(73-1)" Ti( suy minICx) = 72 o Nhir vay ncu dat Xo = OC,, , thi l'{xn) = >/2 = 73-1 rt>;yrtq o.^{^U-wwrt,:-; x = 73 - Ta thu l a i kc't quii trcn ' xeR iL ie uO nT hi Da iH oc 01 / Nhif vay la eo = 1+ x = Xd = 73 - Cac ban Ihijr x c m eo the giai bai toan trcn bang each khac ma l a i dOn giiin hdn hai each tren khong? Nhqn xet: \ ' X|) C O the linh nhif sau: ,c Bai Cho x, y la cac so thiTc thoa man dieu k i c n < x + y + > DifcJng thang qua A B eo phlfdng trinh y+ ] X - r^! , 73 - i n ! m-^X ^m , • nhui R6 rang phu'rtng phap ihj to ro day uy life eiia no qua RJi giai trcn < 0, k h i ta c6 om X ••:(*>Y i " ( - x ) - V x ^ + X + + Vx^ + N/3X + I " (**) bo V i the chi quan l a m den m i n f ( x ) , ta chi can xet x > I S noi tren chinh la giac A B C j>f K h i ta eo: I'M ^ MA+ MC > AC = sfl ^ Thco dinh ! i ham so sin A M O C , ta c6 ba max ' ^ i ! n.f OM, sin 75" sin 45" max OM' MeAABC = max O A ^ ; OB^; OC^} = max {20; 16; 4} = 20 = O A ^ , minP = O M ^ = O H ^ (d day ta ke O H B C ) " '" -'t' " MeAABC Theo cong thi?e linh khoang each tif O den difdng thang y - 2x - = 0, ta c6 oh j i o i :)i.n,} M OC Ro rang (x + y ) = (x;y)eAABC X c l diem M cho M O C = 60"; M O A = 30", v i O M = x f(x) = N/2 M = M„ e A C ca maxP= Do ap dung djnh l i ham so cosin, xet h m h vuong O A B C vd'\A = OC = ' : r diem Ta c(') x ' + y ' = OM^ T a c o r(x) = J x - - - x l c o s " + l - + J x - - x c o s " + l ^ \„i "'^ A = ( - ; 2), B = (0; 4) va C = ( - ; 0) ^ -'-a w ^ ''''Wb ' I \ t : ^ canh), ww I ^' tap y) (kc ce fa Do X < 0, ncn lit (*) va (**) suy IXx) > f ( - x ) thay M(x; lam ok | - { x ) - V x ^ - x + + V x ^ V x + l , •.tjiri ' ^• - - - ''^ kien da cho hdp cac c Ncu ^ c' - phang toa thoa m a n he dieu De /g X c l C c i c h gi£u bang phifdng phap hinh hoc sau day Ta x„ + — = ~ =>xn= yfi 2(1+ V3) s/ Cho y = + y^ Hu0 vJv»- T i m gia t r i Idn nhat va nho nhat cua bieu thiJe P = ' "yf" -x + 2y-8 0; y > { r > > t ^ f < - i 3, t minP = + - m i n O M ^ T i m gia t r i Idn nha't va nho nhat cua b i c u thtfc P = x^ + y^ - 4x - 8y ^ t*M'*'-,:xl'- MeQ Hudiigddngiai a day Q 1^ difdng tron t a m 1(4; 3) Cac d i e m M ( x ; y) thoa man he da cho la toan bo ti? giac A B C D v d i A = (1; 9).B = (0;2), • va ban kinh R = C = (0; 3), D = (9; 0) V i e t lai P diTdi dang: P = (x - 2)^ + (y - Af - De thay m a x O M ^ = M ^ ; m i n O M ^ = M ? 20 Men up '1 1 « om fa M = H x = — ; y = — 2 = 32 24 y =- minP = o M s M , = o \ (*) - (**) ' = -59 Cac ban c6 the tU" nghiem lai cac ket qua (*) va (**) mot each de dang ^Hn xet: Ta co the SIJT dung phiTdng phap lifdng giac hoa de giai bai toan tren " h l 'sau: Tsan- Tiif ( I ) suy '''^i X - = sin , • vdi < (p < 271 y - = 3cos(p T i m gia trj Idn nha't va nho nha't ciia bieu thuTc P = 4x + 3y Tiif P = 4x + 3y = 4(4 + 3sincp) + 3(3 + coscp) = 25 + 12sin(p + 9cos9 Htidiig ddn giai V i e t l a i dieu k i e n da cho diTdi dang (x - 4)^ + (y - 3)^ = 2=10 B a i Cho cac so thiTc x, y thoa man dieu k i e n : x^ + y^ + 16 = 8x + 6y * M = M2 o • X bo B2 = ; y = minP = - — Nhir the maxP = 40 o ce M = D I w o x -A C3 T i r d o suy maxP = 45 o X c cua I tren C D Tir suy maxP = 8^+ 32 = ; P = + ok ^ d day H la hinh chic'u /g m i n M I ^ = H I ^ -' ^' ww i (xem hinh ve) Do = => OM2 = + = 8; O M , = - = ro De thay m a x M I ' = D I ' = 65; : d day M | , M2 Ian li/dt la cac giao diem cua va difdng tron s/ ABCD Ta Gpi I la d i e m I = (2; 4), k h i P = M l ' - 20, d day M ( x ; y) thuoc tiir giac • Men C) (1) (3) Ap dung ba't d^ng thufc quen bie't: V a , ta c6: Tilf d6 suy cac d i e m M ( x ; y) thoa man (1) n ^ m tren diTdng tron tam ta' ~-\/a^ + b^ < a s i n a + b c o s a < Va^ + b^ d i e m 1(4; 3) va ban k i n h R = T a c : - < 12sin(p + 9cos(p< 15 * ^"^^ Cliuyen BDHSG To^n gia tri I6n nha't va gia trj nhd nha't - Phan Huy KhSi Cty TMHH MTV DWH Khang Vlgt Bay gic( liT (3) (4) suy maxP = 40; minP = 10 xet: Ta c6 the giai bai toan tren bang phiTdng phap mien gia trj ham so Ta thu lai ket qua Ucn (vdi phep giai raft gpn gang) X + 3y-10>0 Bai Cho x va y la cac so' thifc thoa man dieu kien • X +y- 6 nhu^sau: : ,•„ ,.v,.;;,r \ iL ie uO nT hi Da iH oc 01 / ' \ ,^ ' : \• / [x-y+2> CO nghiem ' JM.J '•• \ • • !' -y+ = X • f^, j j ^ , ^ j j ~ ' ^ V i l M ' ( l ) ( ) (3) ( ) « , y+ ra-10>0 -y + m - < o y>10 m (6) y>m-6 (7) ''M- y ^ i ^ (8) «^^-^.'^ Ta , ' Ne'um>8thi - m < - m = : > ( ) ( ) o y > - m y>10-m ro i , /g , ' • y m - => (6) (7) o y > 10 - m c i' \r + IS \ • up s/ / 1^-t) (4) x = m - y (5) = m-2y Xet he (6) (7) ta CO / (1) T u f ( l ) s u y r a x = m - 2y • -3y + m + > / fX + 2y = m ^ ^'.^^^'.1 i Goi m la gia tri tiiy y cua P, he ,sau day (an x; y) • ^ ^ "^^ 10 > (2) x +y-6m-6ol0>m>8 / w ^ ; ' Tir suy maxP = 10 o x = 2; y = 4; minP = o x = : l ; y = ^^nh luqn: R6 rang phuTcfng phap giai bang thi va hinh hoc to ro hieu qua Khi m = + 2.3 = Diem cuo'i ciing ma du'dng x + 2y = m gap chinh la diem B(4; 2) Khi m = + 2.4 = 10 Vay maxP =10x = 2;y = 4; P = < = > x = l ; y = AAB*- hcfn h^n so vdi phifdng phap mien gia tri ham so j ^ ^ i Cho X, y la cac so' thiTc thoa man dieu kien sinx + siny = ^ Tim gia tri Idn nha't va nho nha't cua bieu thtfc P = cos2x + cos2y BDHSG Toan gia tr| I6n nhS't vi g\& tr| nh6 nhSit - Phan Huy Hiidng ddn gidi Khai Cty TNHH MTV DWH Khang Vi§t xet: Ta c6 the suT dung phuTdng phap "chicu bien Ihien ham so " de giai bai toan tren nhU'sau: Dat u = sinx; v = siny Khi ta c6 cos2x + cos2y = - 2sin^x + -2sinV = - 2(u^ + v^) Bai loan da cho trd thanh: \'y A-.:.,-':! \ Tim gia tri Idn nha't va nho nha't cua bieu thiJc Q = + 4 :i lit 4 1 smy = V = — u = —;v = l 4sinx = -—;siny = l M s A m m ? = - - = — 2 MsB u = 1; V = — sinx = l;siny = -— w ww MeAB s/ Ta /g ok bo OM^ - O H ^ 4= -1 B om MeAB Ta CO max OM^ - O A ^ = B ^ = + - = - , MeAB u up "\ O c MeAB u ^ ON > OH (1) I c6 the giai each khac l a i hay hdn each giai tren khong? '^^J11 Cho • ' 1 vdi < a < y = t = - - a ^Hn xet: Phu-png phap thj to ro hieu qua viec giai bai l o a n tren Cac hn 10 Cho bon so thiTc x, y, z, t thoa man dieu k i e n x + y + z + t = « • Scr d i CO I d i g i a i nhif vay, v i bai toan cai hon " h i n h h o c " da the hicn 10 qua cdc dieu k i c n x + y = 6; y = t = - - a vdi < a < - bo Tilfdo suy r a m i n P = - 6^2 c L nen ta CO x + y = z + t = - (4) >I nhih itim ^ •••• M•• ~ (4) om Do OMo = 3V2 ; ONo = ro min(MN-) = M o N f , - Bai U'2j Ta tron ddn v j t a i - * Pafu b^ng (2) xay t) thuoc dirdng 6, (2) iL ie uO nT hi Da iH oc 01 / vdi • ' ^6 rang A A B C ton t a i v i thoa " l a n cac tien dc vc dp d a i canh ^"Ja mot tam giac A OB, Cty TMHH MTV DWH Khang Vigt BDHSG Toan gia tri Idn nhit va gia lii nh6 nhS't - Phan Huy Khii Chuygn PhiTdng trinh (1) xac dinh m i e n < x < Ta CO f (X) = _76^-N/2^ 2 2>/2x 276-X 72x(6-x) X + '">'^" ^"'^ ^^"^ f i m m de phiTOng trinh S V x - l + mVx + = 4\/x^ - c6 nghic icm , nen C O bang bie'n thien sau- HUdng dan gidi f'(x) P»i 3- phiTdng trinh 3Vx - + mVx + = V x ^ ^ - p i c u k i e n de (1) c6 nghla la x > Do x > 1, nen V x T T > 0, vay f(x) (1) /2 , ok () nen dieu k i e n dat x+1 i x+1 ,f £ ^ m nghiem f'(x) ^•""•? if :ji> ' ,:>• nh6 nhat ciaa ham so de gi^i b^i toan day hieu qua: Ldi giai gon gang, sang Ban tha'y the nao vc tinh hieu qua cua hai each giai vijfa tnnh bay! iL ie uO nT hi Da iH oc 01 / Ta hay x6t them cdc each giai khac dc nhan thay cdch giai tren la thich hdp nhS't § sCr D g N G G I A TR! LCJN NHAT V A NHO NHAT C U A HAM SO 2: Bai toan da cho CO dang: D E BIEN LUAN PHUCING TRJNH V A BAT PHl/ONG TRJNH C O Tim m de he 3t - t - - m = (4) 00 t| +t2 0 Dieu xay va chi -{t, +12 > up Hitting ddn giai Vie't lai phiTdng fA'>0 ' y'iU mot n g h i c m thupc d o a n Ta T i m m de a (4) CO nghiem t|, t2 m^ < ti < t2 < I Bai l o a n da cho trd thanh: w ww Ta CO f'(0 = 6l - va CO bang b i e n thien sau: • c 3m + 10>0 : Tur ta c6 ce f(t) = t ^ - t - = m (2) c6 nghiem 0 < sin2x < Dira vao djnh Ii Viet, thi t, + ta = | ; t,t2 = om /g ^ -> l - - s i n ^ x + l - s i n ' ^ x + 2sin2x + m o s i n ^ x - s i n x - = m ( ) -3-m>0 o 10 -Y i ^ ; ; , ^ X f'(x) oA'>0o3m+10>0c:>m>-^ Tir suy - V m + 10 /g Vay he (4) (5) c6 nghiem Nhanxet: Tim m de he sau s • f ;;(.,, l i m f(x) = + « ok bo fa - ( m + l)x + 5m + l = (6) , C O nghiem x>3 • ' (7) X ww He (6) (7) v6 nghiem hai tru"dng help sau: a (6) v6 nghiem A ' < o m ' - 3m < o O < m < B a i T i m m de phiTdng trinh 72x^ - ( m + 4)x + 5m + + - x = c6 nghiem b (6) C O nghiem X i , X2 va X| < X2 < >f ,^ * (2) , ^ ,, [A'>0 (1) X|+X20 Do X | + X2 = 2(m + 1); X|X2 " (8) (9) (8)xayrac::> j ( x , - ) ( X - ) > " (10) lit hi t \ ,1 (11) = 5m + 1, nen 2x^ - 2(m + 4)x + 5m + = (x - 3)^ 275 Cty TrjUII MTV DVVH Khang Vi?t (9)(10)(11)« m^-3m>0 < m hoac m > 5m + l - ( m + l) + o m he (6) (7) c6 nghicm m > Ta thu lai ket qua tren Ban thay the nao? ; V(t + 3)' +V(i ^)^ = Bai Cho phiTdng trinh V - x + -Jl + x - 7(2-x)(2 + x) = m '; t+3+ , HUdng ddn giai ^ f| ' D a t t = V - X +V2 + X ^2() Ta =m up -r+ 2l + = 2m ro Bai toan da cho tn'nhanh: om ok bo Id) w - , ft' • i/2 - < m < 276 (2) ^hqii xet: Thco chung toi khong the c6 phUcJug phiip niio khac lai ddn gian hdn max f(t) = f(2) = ; f(t) - f(272) = V - i ihi ( V a y m < — la c a c g i a I r j c a n t i m c u a l h a m so m Uiiiin^ , 1 o - m > — < = > m < — 4 B a i ( h o phiTctng I r i n h ( l o g , \/x | - l o g , x + m = „ - r ! " ' i - m > m i n 1(1) i-4 n,2 o h o a c la phiTdng I r i n h 1(1) = m c6 n g h i e m I r e n D, ( d d a y D i = { t : t < - } c6n D2 = { l : t > } ) 2V2 m - l * om • /g A^/ia/i xet: v d i D = D, u D fa , ft^+mt + 2m-2=-0 T i m m de he ^2 w (6) CO nghiem Trirdc het ta t i m m de he (5) (6) vo nghiem — m>4^-2^/2 , j Ta thu l a i ket qua tren Ban doc tiT danh gia ve tinh hieu qua cua tijrng j ,= »ft ' ^' ' X6t mot bai toan tiftfng tiT sau: Cho phi/dng trinh — — sin ^ ^ B ^ + m(tan x + cot x) - = X T i m m de phu'dng trinh c6 nghiem 3(tan^ + col^x) + m(tanx + cotx) + = « •'•/ft (8) 3(tanx + cotx)^ + m(tanx + cotx) - = )at t = tanx + cotx => t tan x + c o t x tanx cotx (do tanxcotx = > 0) >2 • rtr (8) 3t^ + mt - = mt = - t \ c> 4-3t^ t He (5) (6) v6 nghiem hai triTdng hdp sau: a (5) v6 n g h i c m A = m ^ - m + < o - lyfl m < 3[(tanx + cotx)^ - 2] + m(tanx + cotx) + = ww (5) (iVI'*^'?" [m>-4 DiTa phifdng trinh da cho ve dang: nghiem tren D , , hoSc la phiTdng trinh f(x) = m c6 nghiem tren D2 B a i toan c6 dang m d i sau day: o phiTdng phap t r e n ! R6 rang he c6 nghiem k h i va chi k h i hoac la phiTdng trinh f(x) = m c6 X e t each g i a i khac cho bai toan tren ! f I S t =v Lc(i gidi nhU sau: bo xeD ce cho tru'dng hdp he ok c Trong bai tap tren ta da mcl rong ke't qua suT dung gia t r i Idn nhat va nho nhat 'f(x)-m I' up ,' ro nghiem h o a c la phu"dng t r i n h f ( t ) = m c6 n g h i e m t r e n Dj m >4+ *fe ,! V a y he (5) (6) vo n g h i c m - ^ < m < + 2>/2, ttfc la he (5) (6) co f ( t ) = - c o Ta ;:iYVrt» o m i n f ( t ) = S cosx = - => V T = vo l i ) '•SI-J'" sinx = => sin2x K"? i Do dU'a phufdng trinh da cho ve diing tuTdng du'dng sau: ( l + sin2x) (1 + cosx)^ = m D = {x: + 2cosx > 0, + 2sinx > 0} \ 1\i sinx > - — va cosx > -— 2 n 27t suy — < x < — ^ (1) Vay D = {x: X Ihoa man (1)) Ta X ' 2t — t^ D a t tan — = t A p dung cong thuTc sinx = vk cosx = k h i l + l^ 1+t• 1-t^^ = m o -({^ - f ^ + t ^ + t + l)==m ^ /g n6n ( ) o 1-t^ om 1+ up 2t -lQVte X tf4 Q SR m+24>0om>6 , s/ up /g om c ok 1; bo -4 Vx G [-4; 6] Xet cac each giai khiic niJa nhuT sau: ' Theo bat ding thufc Cdsi, vdi moi x e [-4; 6] tW Cdch 2: Da11 = V(x + 4)(6-x) = V-x^ +2x + 24 (•1.' Xet g(x) = - x ' + 2x + 24 vdi - < x < => g'(x) = - x + 2, va cd bang bic" thicn sau: Tiirdd suy m > thi V(x + 4)(6-x) la cac gia tri can tim cua tham so m + g'(x) '^o rang each giai cung v6 ciing ddn gian va sang sua! f g(x) %h 4: (PhircJng phap thi) ^ ^ ^ ^ y = 7(x + ) ( - x ) , thi ta cd y > va cd: '? ^ 286 I ChuySn dg BDHSG Toan gia tr| lOn nha't Cty TNHH MTV D W H Khang Vi$t gia tr| nh6 nhat - Phan Huy Khii V-x^ +2x + Xet g(x) = -x^ + 2x + vdi - y >0 y >() Pat I = - x ^ + x + 24 = y^ ( x - l ) ^ + y ^ =25 Ta CO g'(x) = - x + 2, va c6 bang bien thien sau: tarn tai diem 1(1; 0) va ban kinh R = : + ^ max g(x) = g(l) = 3; ^ ^ iL ie uO nT hi Da iH oc 01 / Taco true do'i xiJng ^ ' f -2 12m^ - 24m dung vdi Viet lai bat phu'dng trinh da eho du'di dang ( x ^ - x - 8)+ 4V-x^+2X + 8-10 > - m ^ 3cos''x - 20cos^x + 36cos\ 12m^ - 24m moi X e [-2; ] (3) () m ^ - m (2) -i2 • •' ^ ^ ^ ^ f(t) = f(0) = , -i•>j u r^, loan da cho Cdc ban CO tin dieu khon^y Bai Cho bat phuTdng trinh sin3x + msin2x + 3sinx > ,i Ta s/ D a t f ( x ) = Vx + l - V - X v d i - < x < 2>/x+T > Vx e (-1;4) 2^/4^ Tir suy bang bien thien sau: Vay max f(x) = f(4) = Vs max -l m c6 nghiem m < \/5 16 rang each giai vifa phuTc tap, vifa khong sang sua nhi/cach giai bang Nhan xet: Chac chan day la phuTdng phap hieu qua va ddn gian nha't de giai bai f(x) = V x + T - V - x > m jV I Do m > \/5 , nen - m^ < vay (4) v6 nghiem => he (3) (4) v6 nghiem Tiir(2)(3)tac6 m ^ - m < o < m < Tim m de he (3) - m ^ >27(x + l ) ( - x ) (4) iL ie uO nT hi Da iH oc 01 / sail:• • • Taco f'(t) = 21^-1 va CO bang bic'n thien sau phifdng trinh c6 nghiem (vi it nha't x = la nghiem) * X t k h i m > >/5.Tac6 291 Chuy6n 6i BDHSG To^n giA tr| Idn nha't glA tr| nh6 nh^t - Phan Huy Khai 0 f'(t) Vay f(t) ^ ^ ^ f'(x) f(x) ^ = 2%/2 f(t) - f + ' (5) t V I -00 max f ( x ) = Tiif ta suy 0 m t + > VI t e [ I ; 2] OK: ^B^'"*^ '^^"'^ Ta c6 f ' ( t ) = t^+2t + f(t) — + l m (1) f(t) = t+i C O nghiem l[...]... f(t) = max {f(0); f(l)} = max{-3; - 2} = - 2 o0 ^.^ -t^ - 4 t - 2 I f(l) (3) iL ie uO nT hi Da iH oc 01 / Ta = m PWH >2 nen (8) t = m < m < 4 + 272 b (5) CO n g h i c m t| < Ij va khong thoa man (5), tiJc la - 2 < l | < tj < 2 (7) 281 Chuyfin... (vi khi do it nha't x = 4 la nghiem) * X 6 t k h i m > >/5.Tac6 291 Chuy6n 6i BDHSG To^n giA tr| Idn nha't glA tr| nh6 nh^t - Phan Huy Khai 0 0 1 f'(t) Vay 0 1 f(t) 1 ^ ^ ^ f'(x) f(x) 1 ^ = 2%/2 min f(t) - f 1 + 2 ' (5) t 1 V I -00 max f ( x ) = 1 Tiif do ta suy ra 0 A B = sjy + 3 ' = 3 V2 ^ :l,:o;>... trinh da cho co nghiem tren D => dpcm - • 263 Cty TNHH MTV DWH Khang Vigt Chuy6n dg BOHSG Toati gia tii Idn nha't va gia trj nho nha't - Phan Huy KhJIi 2 Gia sur he da cho Ro rang c6 n g h i c m , ttfc la ion tai x„ e D sao cho f(X(,) > X a m a x f ( x ) > IXx,,) > a r'(x) xeD Dao l a i gia sur m a x f ( x ) > a i xeD Gia thie't phan chi?ng he da cho vo nghiem, ttfc la f(x) < a V x e D •a' TO T i j rdo... tai M ( l ; 5) Bai toan da eho CO dang: • -2 ... kien dc he (4) (5) v6 nghiem i He (4) (5) v6 nghiem hai triTcing hPp sau: 07^ Cty TIMHH MTV DVVH Khang Vi$t ChuySn 6i BDHSG Toan gJA t r j dn nhaft vA g& M nh6 nhSt - Phan Huy KhSi fx>3 U)... t = m < m < + 272 b (5) CO n g h i c m t| < Ij va khong thoa man (5), tiJc la - < l | < tj < (7) 281 Chuyfin dfi BDHSG Join gia tri Idn nhat va gi trj nhd nhat - Phan Huy KhSi l(t) = B ^ i... IVITV DVVH Khang Vi$t Chuy§n dS BDHSG Toan gia trj Idn nhat va gia tri nhd nha't - Phan Huy Kh^i + >/5