GIÁ TRỊ LỚN NHẤT NHỎ NHẤT

Cty TNHH MTV D W H ChuySn dg BDHSG ToAn gii trj Idn nhft va g\& trj nh6 nhSX - Phan Huy Khai Cpng tiifng v6' ba bS't d i n g thtfc tren va c6 Tit (7) (8) va ket hdp vdi x + y + z = 1, ta c6 2 2 + > 3[(xy)^ + (yz)3 + (zx)3 2 (xy)3 +(yz)3 +(zx)3 | x N / ? + y > / ? + z ^ J (9) i Dau b^ng (10) xay dong thcfi c6 dau b^ng (6) (9) xy + yz + z x < iL ie uO nT hi Da iH oc 01 / VI => x ^ ^ 2' v^s.; \i x + 2y^ -= X - 2xy'* ' • = 3y^ ^ (2) up ro 3y^^x ok bo (4) (5) ce • ' • , • > , Z + +r x + ' (6) *' xy + y isfl tti - T i r ( l ) (2) suy —^ >x + l - - ^ 'J—^x + l-——- y- + l ?y Da'u bang (3) xay r a o y = 1" .(purr T.v/ix^T (x + l)y" , (3) j, - — ^ > y + - Zillti: ^ (4) ^^'.>,,l_iilii (5) ' Da'u b^ng (4) (5) ti/dng i?ng xay o z = 1; x = r Cong ttrng c-(3) (4) (5) va c6 P > + ^ ^ V ^ ^ - ( ^ y ^ ^ ^ ) x + xz + xz > 3x \/z^ O X = y = z = y + yx + yx > y ^ Do = (x = y + z)' > 3(xy + yz + zx) => xy + yz + zx < }S' (6) Da'u bang (6) xay dong Ihdi c6 da'u b^ng (3) (4) (5) Lai ap dung bat ddng thtfc Cosi, ta c6 ^^'•-.'y.^ + (2) x^ + X = y = z = y +^ + y^ +1 z^ + x^ Thco ba'tdangthiJcCosi, taco y ^ + >2y + hayP>3-^(z^ +x ^ + y ^ ) 3\ Dau bhng (6) xay o dong thcJi c6 dau b^ng (2) (3) (4) (1) w ww + + HUdng ddn gidi , x+1 , (x + l)y^ Taco:— =x+l - ' , ^ Lap lua n tiTdng tU", ta c6 Dau bang (4) (5) tiTdng iVng xiiy y = /;\ = x^ Cpng tirng ve (3) (4) (5) va co P > (x + y + z ) - ^Uy[7 X ~- - Da'u bang (2) xay y = fa •>z x\'z c ' Dau bang (3) xay x = y\ 2 I — TiTdng tir, CO — — r - > y - -z^y^ , y + 2z3 z + 2x- (3) /g X + 2y'' (• ' •: om T i r ( l ) ( ) CO 2xy 3rT -4n= = x - - y V x , Tim gia tn nho nhat cua bieu thifc P = Dau bkng (2) xay o x = y\ >x rr^ ""^-'.V' Thco bat dang thiJc Cosi, thi x + 2y' = x + y ' + y ' > ^ / ^ Bai Cho x, y, z la ba so' du'dng va x + y + z = (1) x + 2y- (9) (lO) Ta x' , _ ,j Vay minP = X = y = z = s/ Ta CO : o d6ng thdi c6 da'u bang (6), (9) o x = y = z = HUcIng ddn gidi / < xay I z + 2x^ y + lz^ + De thay da'u bhng (9) xay o x = y = z = 1, nen da'u bang (10) Bai Cho x, y z la ba so Ihifc dufdng va thoa man dieu kien x + y + z = x + 2y^ + Bay gic( tiT (6) va (9) di den P > Nhdn xet: Trong bai tap trcn ta da suf dung k l thuat Cosi ngiTdc! Tim gia trj nho nha't cua bicu thuTc P = (g) TiJf (7) (8) va X + y + z = 3, suy + 2.3 > |x\/z^ + yyfyi^ + z^/y^ ? X = y = z = o x = y = z = (7) Lai = (x + y + z)^ > 3(xy + yz + zx) (10) Tir (6) (9) di den P > - - => P > T6mlaitadidenminP=l Khang Vi$t , z + zy + zy > ' ' ' 3z^ 117 ChuySn 6i BDHSG Toan gii tr| Ifln nhgt Cty TNHH MTV DWH Khang Vigt g\i trj nh6 nhS't - Phan Huy KhSi Da'u bkng (7) xay o X = y = z = Do X + y + z = 3, nen tir (6) (7) di den P > Dafu bkng (8) xay o dong thdi c6 dau bang (6) (7) x = y = z = l Vay minP = 3x = y = z = Bai Cho x, y, z la ba so thtfc di^dng va x + y + z = Tim gid tri nho nhat cua bieu thiirc P = -J-— + -^r^— + x^+1 y ' + l z/ + l HUdng ddn gidi (1) Ta c6: = - x ^ + I x^+l Theo ba't dang thuTc Cosi, t h i x^ + > 2x (2) Dau bang (2) xay x = (8) TlM GIA TR! L N NHAT VA NHO NHAT H A M SO iL ie uO nT hi Da iH oc 01 / Phrfcfng phap suT dung bat ding thiJc Bunhiacopski cung la mot nhifng phi/rfng phap cd ban dc tim gia tri Idn nha't va nho nha't ham so' (cung nhuf de chtfng minh bat dang thuTc noi chung) ; Giong nhir suf dung ba't ding thuTc Cosi, de c6 t h e ap dung mot cdch h i e u qua phiTdng phap nay, moi bai toan cu the can lifa chon mot each thich hdp hai bo so' roi ap dung bat dang thtfc Bunhiacopski cho hai bo so Chu y r^ng hai bp so can liTa chon khong doi hoi tinh khong am cua cac so hang „„ Bai (Bat dang thuTc Svcic xd) ' ' * Cho a i , a2, a^; b|, b2, bi bi, b2, bj la ba so'diTdng ChuTng minh a? , , a.^ ^ ( a + a ^ + a , ) ^ ' ' Ta up s/ (4) y i+1- l - i ro (5) ww w fa ce bo ok c om /g z^ + l Cong tirng ve (3) (4) (5) va c6 P > - x + y2 + z = -23 (do X + y + z = 3) Tif suy minP = o x = y = z = I Nhan xet: Neu ap dung tiT dau bat ding thufc Cosi x^ + > 2x; y^ + > 2y; z^ + > 2z, ta co: P < - 1 (6) —+ —+ l,x y z Ap dung baft dang thufc Cosi ccf ban, ta co (x + y + z) —+ —+ - > Tur (6) (7) ta CO P < - ( \— +n— + (7) (do X + y + z = 3) >1 BUNHIACOPSKI (3) Tir(l)(2)suyra - - ^ > - ^ = - J x^+1 2x Dau bang (3) xay x = Hoan toan tufdng ivt, ta co — =>- +- +- > x y z §2 PHLfdNG PHAP SCT DUNG BAT D A N G THLfC (8) U yz R6 rang i\i (8) khong suy ket luan gi ve moi lien he giffa P va ^ Phep siJf dung phiTdng phap Cosi ngifcJc la co hieu qua ro r^t bai toan 7, cung nhiT cdc bai todn ttf - ^1 1 ^2 > , > b| + b2 + bj ^.1 [• ,.„ ,.t Hiidng ddn gidi - a^ ^ thuTc va Bunhiacop.ski v d i hai day Ap diing32ba't+, dang Vbi" 7^7 („2 „2 I „2^ hj b (b, + b + b - , ) > ( a , Ta co: b| t +32+33) r> u L L ^ , a? a^ a? ( a + a j + a , ) ^ Do bi + b2 + hi > 0, nen co: -1- + _i + _2 > }LJ £ -IL ^ suy dpcm b| b2 bj b| + b2 + bj «! a2 Vb, 7b2 n-' - o \ P i = Dau bang xay = AJ a, — = a, — = a, b, b2 b3 tfx ^h^n xet: DSc biet la'y bi = aid > (i = 1, 2, 3) ta co ket qua sau: Cho so a,, 32, 3 va Ci, C2, C 3iCi > Vi = 1, 2, vu- - 3? -AI a? Khi ta co: —L- + — ^ + a|C| a2C2 (31+32+33)^ > 33C3 V — | C | + 32C2 — x + a3C3 Cho ba cua so' thifc Tim gia X,tri y,Idnz lanha't bieudiTdng thtfc Pthoa = ^^—-^ xman + +dieu y + 1kien: +z-+—xyz 1- = ; IQ Chuyen 66 BDHSG Toan gii trj lOn nhift va Q\A trj nh6 nhft - Phan Huy Kh5i Cty TNHH MTV DWH Khang Vi^t Xa c : V i e t l a i P diTdi dang sau: P = - r x +l + " y+1 L u c nay: -+ -+ =^ + Y vi't^'^^'-r - -^97 Z x^+y l'f*>- - ' ' X Y P S ; Z z^ x^ Y Z + Z^ y+1 ^ :j ^ (X + Y + Z)^ ^ (X + Y + ^J Zf , (3) up T i f theo (1) suy ra: P < * c om B a i Cho ba so' ihuTc diTcJng x, y, z va x + y + z = /g V a y max P = o x = y = z = ro D a u b i n g t r o n g ( ) x a y r a < = > X = Y = Z o x = y = z = P>-^— j + x'' + y ' ' + Z ' ' xy + yz + zx rr (x + y + z)"^ + \ xy + yz + zx (1 + + 1)2 xy + yz + zx xy + yz + zx =9 x + y + z + x y + 2yz + 2zx * + - i ^ l ,u> A xy + yz + zx (3) ' \ (4) , , •••! • • x=y=z= - Bai Cho x, y , z la cac so'thU'c di/dng thoa man dieu k i e n : x y z = T i m gia t r i nho nhat cua bieu thtfc P = - r - ^ ^ + +-r-^ x ' ( y + z) y ' ( z + l ) z-\ + y ) I jXf: (1) i A p dung bat d^ng thuTc Sv^c-xd tuT (1) c6: 1^' 'I • —+ — + P> (2) X y - (xy + yz + zx)2 z 2(xy + yz + zx) 2.2 (2) 2(xy + y z + zx)x^y^z Do xyz = 1, nen tir (2) ta co: P > ^y + y^- + ^^ DSfu b i n g (3) xay x = y = z = D a u b i n g (2) xay r a o x = y = z = j + - Nhif vay m i n P = 30 Gia t r i nho nha't dat di/dc k h i x = y = z = ^ ok :'' D a u b i n g (1) xay r a o x = y = z = - Tiir(l)fac6: , x y + yz + zx :2 ce fa , xy + yz + zx v2 12 Tac6:P = — + +—^ x ( y + z) y(z + l ) z(x + y ) w fi ww x+y+z + Hudngddngidi bo xyz Ta c6: xy + yz + zx = xyz — + — + - > xyz = 9xyz X y z ] x+y+z xyz +z TO (3), (4) suy P > 30 v i dau b i n g xay o Z + " X Y + Y + Y Z + Z ^ + Z X + X ^ ~ ( X + Y + Z)^ T i m gia t r i nho nha't cua bieu thiJc P = - r — \ ^ x^+y^+z^ , Ta x+l xy + yz + zx Dau bang (4) xay o X = y = z = - s/ Bunhiacopski) ta c6: ^ T+ =:>xy+ yz + zx < - = > >21 xy + yz + zx 2ZX + X ^ A p dung bat dang thiJc Svac-xd (Bai - dang dac biet cua baft dang thiJc ^ + y + z L a i c6: = ( X + y + zf = x^ + y^ + z^ + 2(xy + yz + zx) " 2X + Y ^ Y + Z ^ Z + X 2XY + Y^ =x DSu b i n g (3) xay o x = y = z X ^ xy + yz + zx iL ie uO nT hi Da iH oc 01 / - + jji^vH X p dung baft ddng thuTc Svdc-xO ta c6: Y>0,Z>0(vlrorang -pi/iq , P + y1^ + z xyz = 8) z + l 9Y 9Y 77 Datx= — ; y= — ; z -—.Khid6tac6:X>0, Y Z X , ' v*- L a i theo bat d^ng thtfc Co Si ta c6: xy + yz + zx >3^{\yzf • i> ; =3 • • (3) (4) (do xyz = ) • • " • ' T Cty TNHH MTV DWH Khang Vigt ChuySn dg BDHSG To^n gii tri lOn nha't va gia tr| nh6 nhS't - Phan Huy Kh^i Dau bang (4) xay X = y = z = Ta c6: |,4 y.^^r^—^^ ; Da'u bang (5) xay dong thdi c6 dau bang (3)(4) o x = y = z = .:• V a y m i n P = ^x = y = z = - ' v ^• Bai Cho x, y, z la ba so thiTc diTcJng thoa man dieu kien: xyz = X y z Tim gia trj nho nhat cija bieu ihuTc P = - + + • , , 2' M 1+ - 1+1+ iX\ z X * " i > (3) X w « i Bai Cho x, y, z la cac so thifc difdng ihoa man dieu kien: xy + yz + zx = Ap dung bat dang thuTc Bunhiascopki cho hai day s6': • ^: x ^ y ^ z ^ 1,1,1 • y Z xy + yz + zx = X , y, z cung dau = xy c ^ < o x=y=z= -2N/3 Tit suy P = 2v^ „ 2V3 B i i Cho X , y, z la cac so thuTc diTdng thoa man dieu kien: x + y + z = x-^ y+z Hudngddngidi = y = z = Huditg dan giai y = xz 16 • '•'•^'^ a- (4) i t Tim gia tri nho nha't ciia bieu thiJc P = x"* + y"* + z* ^ X Tim gia tri nho nha't cua bieu thtfc P = ww o X = Y = Zx = y = z = l x^ = yz 2V3 Ta s/ up ro /g om bo ce ' • Dau bang (4) xay dong thdi CO dau bkng (2)(3) Vay P = y Gia tri nlio nha't dat difdc x = y = z = — j - hoSc l a x = y = z = - - ^ fa X = Y = Z Tiif (2), (3) co: P > X, (2) ok ~ 2(XY + Y Z + ZX)2 D e t h a y r k n g ( X + Y + Z ) ' > ( X Y + YZ + ZX) ' z, Dafu b^ng (3) xay va chi c (X + Y + Z)^ Dau bkng (2) xay o X = Y = Z Dau bang (3) xay y, z lflf(l),(z;suyrax +y +z XZ+2YZ ' ' Ap dung bat d^ng ihufc Svac-xd, ta c6: P > X, z ' (chu y la xyz = 1, ncn c6 the ddi bien nhuTtren) Z X Y X Y Z Y Z + ^ • + , 2Z , 2X Y + 2Z Z + 2X + -X + 2Y 1+— 1+ — X i Y Z Y^ X2 ± (1) orfJ.- + ZY+2XY y^ + z ^ f ^ 3(x^ + y'* + z'*) > (x^ + y2 + z^ f H- x y z Xet phep ddi bien x = —; y - — va z = —, vdi X, Y, Z > Y Z X XY+2XZ ) ( l ^ +1^ +1^ ) > (x^ Taco: ( x ^ + y ^ + z ^ ) ( z ^ + x + y ) > ( x y + yz + zx)^ = > ( x ^ + y ^ + z ^ ) > ! HUdngdangiai = + Dau bling (1) xay o x^ = y^ = z l Lai ap dung baft d i n g thiJc Bunhiascopki cho hai day so: J ^ ' + (5) iL ie uO nT hi Da iH oc 01 / Tir (3), (4) suy P > I , + —— z+x +• x+y -Jwrm^ z^ x"* v'* z"* Tac6: p = J— + ^ +^ =— +^ +— (1) y + z z + x x + y xy + xz yz + xy xz + yz (x^+y^+z^f Ap dung bat d i n g thuTc Svac-xd, ta c6: P > ^ ' (2) 2(xy + yz + zx) Da'u bkng (2) xay rax = y = z = l (dox + y + z = 3) D l tha'y r^ng: x^ + y^ + z^ > xy + yz + zx (3) Da'u bkng (3) o X = y = z = 123 Chuy6n dj BDHSG Toan g\& tr| lOn nha't T i r ( ) , ( ) t a c : P>^5^ ^ Uiin nhien ta c6: + + Cty TNHH MTV DWH Khang Vi^t g i i trj nh6 nhgt - Phan Huy Kh5i - > V a y m i n P = x = y = z = ^ (4) ^^ Cac ban hay so sanh cdch giai v d i cdch dung phiTdng phap them bdt (5) hang t ^ va bat dang thiirc Cosi Dafu b^ng (5) X = y = z = (6) iL ie uO nT hi Da iH oc 01 / T i m gid trj nho nhat cua bieu thtfc P = Dau bkng (6) xay o dong thdi c6 dau b^ng (4), (5) X d p dung bat d i n g thiJc X (* X, y, z > VI X bo o • ,< ^ \2 = J 2 ^\ i (4) X = y =z = ww fiai Cho x > 0, y > 0, z > va x + y + z = x^ x+y z^ j +— —+ y+z z+x T i m gia tri \dn nhat cua bieu thi?c P = X^ +zf A p dung ba't d^ng thuTc Svdc-xd, ta c6: P > ^ ^ ^ / ^ ^ ^ x + +z = \ y+2 ^ V i e t l a i P duTdi dang sau: p= I x-''+2xy 4 y^+2yV y2 ^2 r- + — - — + x + 2y^ y + 2z^ z + 2x^ HUdng dan giai L^i giai nhu sau: (\+ (0 (2) ^ Vay m m P = - o x = y = z = — ^ + y + z = T i m gia t r i nho nhat cua bieu thiJc P = • DSu bSng (4) x a y o ddng th6i c6 dau b i n g (2), (3) Tilf d6 suy m i n P = ^ o x = y = z = Ta thu l a i k e t qua tren Cho I 6(x2+y2+z2) fa 2' tlci-ivi';::!;'• M d c khac ta c6 x y + yz + zx < x^ + y^ + 7? ce _3 Ta c6 b a i toan tuTdng tif sau: x^+y^+z^) Tit(2), (3) ta c6: P > - L —J- w Ta s/ up D o x^ + y^ + z^ > x y + yz + zx, nen tif (*) suy ra: c > x^ + y^ + z^ ok om /g >z2 Cong tuTng ve bat dang thtfc tren ta c6: P + — — ^ ^ z + 2x + 3y ' + y + z + 5(xy + yz + zx) Dau b i n g ( ) o x = y = z = ro x^+y^+z^ ^ ( x + y + zr ' DSu b^ng (2) xay r a o x = y = z = , y(z + x ) x +y 1— p> Cosi de giai b ^ i toan tren nhtf sau: x(y + z) x^ Taco: - ^ + " ' ^ " " , y+z z(x + y ) ^ • ^ + 2y + 3z y + 2z + 3x X Ap dung baft d^ng thufc Svdc-xd, ta c6: v Ta CO the suT dung phiTdng phap them bdt hang tuT Z + X s HO ' T a c : P = ^—-^^^ ^ - +- — T ^ + T T"X + x y + 3xz y + y z + 3xy z + x z + 3yz ^^ajV ¥?f - ' m (i;xrri 4 Vay m i n P = - o x = y = z = ' ' Hudng ddn giai = y = z = Nh^nxet: ' " " • ' '' p ^ l Cho X > 0, y > 0, z > va x^ + y^ + z^ = Tilfx + y + z = ( ) , ( ) s u y r a : P > | o ^ z^+2xV (3) BDHSG Toan gii trj Idn nhSt va gia tr| nhd nhat - Phan Huy KhSi Cty TNHH MTV DWH Khang Vi$t Theo baft dang thtfc Svac-xct, ta c6: , Dau bang (1) xay H'- i'sifn V > ••• \ -f Da'u bang (1) xay o x = y = z = (do x + y + z = 3) x^+y'*+z^+2(xy+yV+zV) Tac6VP(l)= , , , ) , 2 2 , b c ax bx cz ,, x y 1 z Ti/dng t y co: ; >p —+ —+ - = X y z I b c a' —+ —+ X y z I c a b —+ —+ — X y z bx + cy + az (2) A p dung baft dang thtfc Cosi ta c6: x'* + x"* + x > 3x^ a iL ie uO nT hi Da iH oc 01 / Chuy6n cx + ay + bz x = y = z = (3) Dafu b^ng (2), (3) deu xay o x = y = z = >33zz' \ ^ — ax + by + cz Cong tijfng ve bat dang thuTc tren ta c6 ( x ^ + y ' * + z ^ ) + (x + y + z ) > ( x - U y U z ^ ) * y H l + l>3y - \ infid v .1 , up ok ww B a i 10 Gia suf x, y , z la cac so thiTc di/cfng cho - + - + - = X y z ' Cho a, b , c la ba h^ng so difdng cho triTdc T i m gia t r i Idn nhat cua bieu thiJc P= I + ax + by + cz bx + cy + az c^ (a + b + c)^ ax bx cz ax + by + cz ^ ^a b c —+ —+ ax + by + cz ^x y z, 1 P fl P + - + - + c —+ —+ U y z; Ix y 'L) ax + by + cz bx + cy + az 1 cx + ay + bz a+b+c DSu bkng (4) xay o dong thdi c6 dau b i n g (1), (2), (3) O x = y = z = max ? o x = y = z = a + b+ c x + y + 6z 3x + 6y + 2z = 6x + 2y + 3z {St' n^unhir - + - + - = v i x > , y > , z > X y z rim 11 Cho X, y, z la cdc so diTdng ^ y + 2z z + 2x -+• x + 2y Hudng ddn giai D i l t S = a + b + c > A p dung bat d^ng thi?c Svdc-xd, ta c6: b^ ' + bf -l T i m gia trj Idn nhS't ciia bieu thuTc P = cx + ay + bz HUdng ddn gidi a^ cx + ay + bz Ap dung: N e u a = 2, b = 3, c = ta c6 ket qua sau: Olm^h w V a y P = X = y = z = ' (6) fa Dafu bkng (7) xay de tha'y o x = y = z = l bo Tiir(l),(6)suyraP>l / i ' " V$ymaxP= ce Ta s/ (4) Tir (3), (4) suy ra: x" + y^ + z" > x U y ' + z\ s,\) T u r ( ) , ( ) c : V P ( ) > bx + cy + az + c ' , om Do X + y + z = 3, nen c6: x^ + y^ + z' + 2(x + y + z) > 3(x + y + z) ro 3|di'a/' /g Tijf suy ra: x ' + y^ + z^ + > 3(x + y + z) '"'^ + De thay dau b^ng (5) o x = y = z = 1 Tif gia thiet - + - + - = \a S = a + b + c suy ra: X y z + + >3z * + (\ r a —+—+U y (3) Theo bat d^ng thuTc Cosi ta lai c6: x^ + + > 3x =>x^ + y^ + z ^ > x + y + z (1) ' i ^ t l a i P dirdi dang sau: P = — - — + — ^ — — + — - — xy + 2xz yz + 2xy xz + 2yz - (1) va theo ba't d i n g thtfc Svac-xd, ta c6: (1) Chuy6n P> BDHSG Join gii trj Idn nha't va gia trj nh6 nh3"t - Phan Huy Khii (x + y + z ) ! hayP> x y + 2xz + yz + x y + xz + 2yz ^ , ^ Cty TNHH MTV DWH Khang Vigt (^^y^^-)' 3(xy + yz + zx) / = x (2) i Ro rang (x + y + z)^ > 3(xy + yz + zx), nen tijf (2) suy ra: P > Dau b i n g (3) xay Nhdit X Uj'iih •}!;:> 'ixil, ^2y^(l-y^)^(l-y>-^VV^4(l-2y^).l Tacd: = y = z > xet: (4) L a i ap dung ba't dang thiJc Bunhiacopski v d i hai day: d i n g thufc Bunhiacopski iL ie uO nT hi Da iH oc 01 / Cach giai tren la diTa vao bat d^ng thtfc Svac-xd (mpt dang dac bi^t cua bat ' X e t each g i a i sau day diTa vao ba't dang thiJc Cosi Dat X = y + 2z; Y = z + 2x; Z = | X > 0, Y > 0, Z > 4Z + X - Y 4X + Y - Z ; y= Y ; z= 4Z+X-2Y 9Y 9Y 9Y ^/3 4X + Y - Z 9Z Y^ Z^ f z fx —+ + — —+ — +— 9 Iz Ix 9Z j 9Z X^ Zj Z (Y + 3- —+ —+ IZ Y x^ zj (*) hayP>l (**) c om Theo bat dang thurc Cosi, t i i f ( * ) suy ra: P > ^ + ^ + -^ + l - | ok De thay dau bang (**) x a y r a < : : > X = Y = Z > O o x = y = z > = y = z bo Vay P = l o x ce D o ro r^ng phiTdng phap giai sijT dung bat dang thuTc Svac-xcf la gon gang w B a i 12 Cho x, y, z thoa man d i c u k i e n : x^ + y^ + z^ = T i m gia t r i Idn nha't cua bieu thiJc P = xy + yz + 2zx Hudng ddn giai •xy + y z < |x^ ^ +z^^ De thay: 2zx < x^ + z^ = - (2) P Ta 4Y + Z - X 9X s/ vax= talaico: up 4Y + Z - X 9X^9 ,jr ro CO 2y X + /g K h i ta „ Z = X (3) De thay dafu bkng (3) xay x = y = z Vay P = o X _ y (3) l + 2z^ l + 2x-^" Hudng ddn giai , Theo bat dang thii'c Svac-xd, ta c6: P > ^ ^ ^ y + ^)^ "'3 + ( x ^ + y ^ + z ^ ) C>au bkng (1) xay o /n x = y =z= I Theo bat d i n g thtfc Cosi, ta cd: x^ + + > 3x y^ + + > 3y >'•••'' ^^f>}b "\ •: 'hv f\C\ Chuy6n dg BDHSG Toan g\A tfj lOn nha't Cty TNHH MTV DWH Khang Vi$i gia-trj nh6 nhat - Phan Huy KhSi z ' + + > 3z TiJf va difa vao gia Ihiel x + y + z = suy ra: + + z^ > P = (2) Dau bang (2) xay rax = y = z = l (3) P> Da'u b^ng (3) xay dong thcfi c6 dau bang (1), (2) ,;„.,.:u „vi ,.,::;f, VM'M-.r, B a i 14 Cho x, y, z la cac so' thiTc duTdng T i m gia t n nho nha't cua bieu thtfc y +yz + z :r + -^2 T + 2 x +xy + y z +ZX + X • ' HUdngdangidi ' i ^ X x ^ ( y ^ + y z + z^) y ^ y ^ ( z ^ + z x + x^) z z?(x^ + xy + y ^ ) A p dung ba't dang thuTc Svac-xd, ta c6: 2\ up x^+y^+z^ hayP>^-— /g x^+y^+z* + f x y + y V + z V ) , , , ^ ' (2) , bo Theo bat d i n g t h i J c C s i , t a c : ok 2(x^y^ + y^z^ + z^x^ J + ( x y ) ( y z ) + (xy)(zx) + (yz)(zx) om ro i L x^(y^ + y z + z^] + y^(z^ + z x + x^) + z ^ ( x ^ + x y + y^) c p> , , _ s/ 2 «, ^ ce x ' + y^ + z ' > x y + y V + z V (3) (4) w z'• + z* + z > z ^ ww sfx^y^+yV+z^x^] Tiir (2) (3), (4) suy ra: P > — — ( hay P > , 3(x^y^+yV+z^x''j D c tha'y da'u b i n g (5) xay o ' (5) x = y = z > V a y m i n P = X = y = z > i i a i 15 Cho x, y, z la cdc so thiTc diTdng T i m gia t r i nho nhat cua bieu thiJc „ P,= X yjx^+Syz + ,y z / + / 7y^+8zx ^jz^+8\y HUdng dSn giai (2) -i,;- = (x + y + z ) ( x - ' + y ^ + z ^ + x y z ) ; : :• A p dung bat dang t h u t Cosi, ta co: v^ r < (3) \ (X + y + z ) ' = x ' + y^ + z ' + 3(x + y + z)(xy + yz + zx) - 3xyz ri >x^ + y ' + z^+ 7 x y z ^ / x y ? - x y z v, hay (X + y + z)^ > x^ + y^ + z^ + 24xyz .t* > A ' , - y g „ s f : r X) < x (4) Thay (3), (4) vao (2) va c6: P > ^ ^ i i ^ ^ i ^ = (x + y + z r Dg tha'y dau bling (5) xay o (5) x = y = z = A^Aa/i A:^^; Ta c6 bai tocin tu'dng tu" sau: \.T: «Vt Cho X > 0, y > 0, z > va x + y + z = T i m gia tri nho nha't ciia bieu thtfc P = fa x^y^ + fz^ + z V > (xy)(yz) + (xy)(zx) + (yz)(zx) x7x^ + y z + y^jy^ +8zx + z^z^ + x y (x + y + z) x ( x ' + y z j + y ( y ^ + z x ) + z ( z ^ + x y j V i e t l a i P diTdi dang: (x + y + z ) - V ^ V x ^ x ^ + y z + ^/y.^/y^/y^ +8zx + VZ.N/Z^Z^ + x y J Ta (1) z^z^ + 8xy A p dung ba't dang thuTc Bunhiacopski, ta c6: Vay P = l < = > x = y = z = l P= yyjy^ + 8zx iL ie uO nT hi Da iH oc 01 / = y = Z=l + 8yz Til' (1) va theo bat dang thifc Svac-xd, ta c6: Tird), (2)suyraP>l O X i l l h:l x7x^ x'^+Syz y^+8zx z^+8xy Ta giai nhi/ sau: P = -j-^ + X +8xyz y +8xyz A p dung bat dilng thuTc Svac-xd, ta c6: P > z2 z +8xyz ^ (x + y + z) x' + y +•/: + x y z Theo bai tren ta c6: (x + y + z ) ' > x ' + y^ + z^ + 24xyz (*) (**) (***) Tir (**)^ (***) suy ra: P > ^^^^^^'^^ hay P > ^ = (x + y + z) x+y+z Vay P = l o x = y = z = - ^aj 16 Gia sur x, y, z la ba canh cua mot tam giac c6 chu v i bang 12 V i e t l a i P dtfdi dang sau: Ill ChuySn 6i BDHSG Toan glA trj lOn nha't Cty TNHH MTV DWH Khang Vigt glA trj nh6 nha't - Phan Huy KhSi pai 18 Cho x, y, z la ba so'dtfcfng thoa man dieu kien: , y+ z- x / +x - y x+y - z Tim gia tri nho nhat cua bieii ihiTc P = + + • 3x + y - z 3y + z - X 3z + X - y x(x- l) + y ( y - l) + z(z- 1) < ^ Hiidlng ddn gidi Dat X = y + z - x; Y = z + X - y; Z = X Tim gia tri I6n va nho nhat eua bieu thufc P = x + y + z + y - z HUdng ddn gidi Khi ta CO X > 0, Y > 0, Z > ;y = Z + X ;z= Viet lai gia thie't da cho diTdi dang: X + Y ( x ^ + y ^ + z ) < 3(x + y + z) + iL ie uO nT hi Da iH oc 01 / De thay x = Y + Z Tiif ta c6: P = Ap dung ba't ding thuTc Bunhiacopski, ta c6: Z (X Y + Z ^ Z + X ' ^ X + 2Y X Y + 2XZ Z Y + 2XY X Z + 2YZ' ( X + Y + Z)^ De thay P = ' X = Y = Z o x = y = z = Vay P = Gid tri nho nhat x, y, z la canh cua tarn giac deu co chu vi bang 12 x'; ro HUtUg d&n giai om >— 3+x^ j = x = 11 Tif(l)tac6: x + — + , 2x V X+ - x > 1+ >X + x^ ww 11 + - + 2x 2x (1) + y + z)^ - 3(x+ y + z) - < - l < x+ y+z < o - l < P < o (3) x+y+z=— o x +y+ z= - x =y= z TiTdng tu'P = - l c i > x = y = x = - - Vay max P = o x = y = x = —; P = - l o x = y = x = — sCf DgNG BAT D A N G THLfC D E TIM GIA TR! L6N NHAT VA NHO NHAT CUA HAM SO Phrfring phap xua't phap tuT mpt bat dang thuTc da bie't tijf trtfdfc PhiTdng phap xuat phap ttr mot bat dang thiJc nao dafy da diing, sau bicn =:- + X + X doi bat dang thtfe dang P ^ a (1) (hoae P < a), d day P la bieu thiJc can tlm gia tri nho nha't (hoac \dn nha't) Sau chi phan tur da cho tfng vdi phan tur thi P dat gia tri a, ta se suy ke't luan y Dau bang (3) xtiy o ddng thcli c6 dau bang (1), (2) o x = Vay f(x) = i Đ3 CAC PHadNG PHAP THONG DgNG KHAC c Xy w Dau bang ( ) xay o V bo 1+ 3.1 > ce | l + - ^ > + - => x^ X x'; ok 1+ - V fa ' W - > 3+ (9 + 7) + V X' J o /g Ap dung ba't dang thtjTe Bunhiacopski, ta eo: (2) ' (X Ta 1+ - V up V s/ + 2x > x;, Z^) o Tif suy ra: P = (tinh X = y = z = 4) Bai 17 Cho x > Tim gia tri nho nhat cua ham so l(x) = x + + y + z)^ < 3(x^ + y^ + TiJf ( ) , (2) suy ra: (x + y + z)^ < 3(x+ y + z) + ^, Tilf ( ) va thco bat dang thuTc Svac-xd, ta c6: P > ( X Y + Y Z + Z X ) Do (1) y x = dang thtfc thich hdp vdi de de c6 the bien doi ve ba't dang thiJc dang ( ) Viec lira chpn du'pc tien hanh b^ng each diTa v^o cau true cua bieu thtfc • P ban dau cung nhtf cac gia thiet cua bai toan P^i Cho X, y, z \h ba so diTtfng va thoa man dieu kien xyz = Cty TNHH MTV DWH Khang Vigt Chuyfin dg BDHSG loin g\i trj \6n nhi't vS gia tr| nh6 nhSft - Phan Huy KhSi T i m gia tri Idn nhat ciia bicu thtfc P = — — ^ +—— xUy'+l y^+z^+l + => x* + y'* + 7* > xyz(x + y + z) — ^ z-'+x-Ul Nhan xet dMc ch^ng minh: De tha'y dau bang ( ) xay x = y = z Do xyzt = 1, nen ttr (1) c6: HUdng ddn gidi H i e n nhien ta c6 bat dang thtfc sau: - xy + > xy I (1) * I Dau bkng (1) xay o x = y * H«*' (x + y ) ( x ^ - x y + y ^ ) > x y ( x + y ) => x^ + y ' > xy(x + y) ' \ x ' + y % > xy(x + y + z) Lap luan tiTcMg tiT ta c6: " ' < ! x"* + y"* + xy(x + y + z) i n ; fMJ"'' ^ (3) — 3—- < 7 - — ; y-^ + z-^ + yz(x + y + z) (4) Ta up ( ro ) /g ^^^^^ =— =1 xyz(x + y + z) x y z =y=z= l ce fa "1 P W' vho ca'u true cua bieu thtfc P Bai Cho x, y , z la ba so diTdng va xyz = T i m gia tri Idn nha't cua bieu thiJc 2 X** + y " + z* > xyz(x + y + z) ^^^^ ' 7 y^z^'+y'+z' '^^ 2 7' z^x'^+z'+x' HU^ng ddn gidi rll r.:X.,X'''-':^> u,ul^k; x V + x^ + y^ > x V + xV'(x + y) xV xV Tir (1) co: ^^^^^-^^^^ ' (1) That vay: x U y " + z^ > x V + y^z^ + z^x^ => 2(x^ + y^ + z^) > (x^y^ + y^z^) + (y^z^ + z^x^) + (z^x^ + x^y^) > 2xy^z + 2xyz^ + 2x^yz (1) Dau bling (1) xay o X = y l^zSt^+x^/t^+x^+y'^+r TrU'dc het ta chi?ng minh rang v d i m o i x, y, z ta c6: ( x ^ - y ^ ) ( x ^ - y ' * ) > = o x ^ + y ' > x V ( x + y) ^ fj^^„gj^,^gi^i x''y^ + x ' + y ' Do X, y 1^ cac so di/dng nen ta c6: " w - x ^ + y ^ + z ^ l ^ y ^ + z ^ + t^ + vt.^a i.,MiiMal.b''-5;rvi6 ww ' ' (6) - j ' ' ' ' " " " ' " • ^' B a i Cho x, y , z, I la bon so thiTc dtfdng cho xyzt = I T i m gia trj Idn nhat cua bieu thiirc (5) r *-v i r ' , ( • ^ ^ " Ihtfc (1) bai nhi/ng viec suT dung no cung la le tiT nhien v i diTa JằuH.ằ v)Ơ\m.i ^lAS â M I I G (x - y)^ > i=> x^ - xy + y^ > xy ,,,-;Af.tt, .1; -(4) I ^ A^/iflH xet: Trong bai ta da su" dung bat dang thiJc (1), la tiif bat d i n g thiJc h i c n nhien ^ :/ ; i r -.Hhihkt.Mhlt., Vay max P = l o x = y = z = t = I bo At Way max P = l o x X = y = z = (do xyz = ) i4 U :i 'I :5 m (3) Nh4n xet: day ta difa v^o b5't d i n g thiJc (1) N o khong ddn gian nhiT baft ok *> ?-t.i's/f w f r v ^ , De iha'y dau bang (6) xay r a o x = y = z = t = l c om Dau bang (6) xay o dong thdi c6 dau bang (3), (4), (5) < Dau bang (3), (4), (5) tiTdng iJng xay r a o y = z = t ; z = t = x ; t = x = y zx(x + y + z) c6 P < — Cong tirng ve (2), (3), (4), (5) va chii ^ xyzt = 1, ta c6: P < (1) Da'u bkng (4), (5) liTdng uTng xay k h i y = z; z = z Cong ti^ng ve'(3), (4), (5) (2) y + z +1 +1 yzt(x + y + z +1) t + x + y + txy(x + y + z + t) -(m — xyz(x + y + z + t ) Ztx(X + y + Z + t) s/ z'+x^' + l „ ji-i — - Z4^j4^^4^1 Dau bang (3) xay o dau bang (1) xay o x = y TiTdng tur la c6: L Dau bang (2) xay o x = y = z (2) -, —-< < —— x'* + y"* + z"* + ; D o x y z = l , n e n t i r ( ) c : x ^ + y ' + > x y ( x + y) + xy/ J Do X, y, z, t > 0, nen ta c6: — iL ie uO nT hi Da iH oc 01 / V i x > , y >0,nenlil'(l)c6 + ^ xyz(x + y + z) + xyzt => x'* + y"* + z"* + > xyz(x + y + z + t) x"* + y'* + x^y^+x^+y^ ' '' hay x^y^+x^y-Xx + y ) •' x^y^ + x ^ + y ' ' -'y' x^y^+x^+y^ < ' • ' „ 1 l + xy(x + y ) xyz + xy(x + y ) ' xy(x + y + z) / >^ ^ • (2) 4Vt 135 Chuy6n BDHSG Join gia tr| Idn nhSft gii trj nh6 nhjit - Phan Huy KhSi Cty TNHH MTV DWH Khang Vi$t Bai Cho x, y, z > va thoa man dieu kien x + y + z = Tim gia tri nho nhat cua bieu thtfc Vay P = X, y, z thoa man (6) f^h&n xet: Vice su" dung ba't ding thiJc (2) 1^ dieu phai kheo leo mdi nhan Bai toan la sur ket hdp nhuan nhuycn giiJa phiTOng phap sur dung mpl bat dang thtfc da bie't trifdc vati viec ap dung ba't dang thurc Cosi HUdng ddn giai DiTa vao gia thiet: x + y + z = 2, ta c6: , 2(x-^ + y-^ + Pai Cho x, y, z la cdc so thiTc diTOng va thoa man dieu ki?n ; (*) iL ie uO nT hi Da iH oc 01 / Tac6:2P = + ( x ' * + y ^ + z * ) - ( x ' + y ' + z - ^ ) , •32-3x2=z^ = 16-4y^ ) -(x^ + y^ + z^) = x-\ - x) + y ' ( - y) + z3(2 - z ) = x \ y + z) + y \ + x) + z \ + y) >• • • Dg thSy ta CO b^t d i n g thiJc sau: (1) _ , ^* ' ^ rvs • ' u - - ' ' 16-z^ Tir gia thiet, ta co: y ^ — - •; ' TMtvay: s/ (3) up Do (3) dung VI X > 0; y > 0; z > Vay (2) dung /g ro Dau b^ng (2) xay o xyz = bo + y + z = 2, n6n ta c6: 2(xy + yz + zx) (x^ + y^ + z^ j < =>0 nen (2) o ^ ,;, (x + y)(x + z ) > ( x + 7yz)' z-Sx-^ + y + 7(y + z)(y + x) (X ww x-Uy^ x + 7(x + y)(x + z) A p dung bat dang thiJc Bunhiacopski ta c6: w V a y P = Q T i r d o suy ra: z fa = (x - y ) + (y - z) + (z - x ) = y HUdng ddn giai ok bo x'-y» ^ ce p c Khid6tac6: X n om x ^ + x y + y^ dung bat dang thuTc cho tru'dc (cac bat dang thtfc (2), (4)), cho den suT dung Bai Cho x, y , z 1^ cac so thiTc diTdng T i m gia tri Idn nha't cua bieu thiJc: /g X e t dai liTdng: Q = siJf bS't dang thiJc Cosi diing de giai bai loan t i m gia t r i nho nha't cua P dat ro HUctng ddn giai z + 7(z + x)(z + y) (3) I 2z + ^ x y •, (3) (4) D a u b i n g (3), (4) ti/tJng i^ng xiiy z = x; x = y 14Q Chiiyen dS BDHSG Toan g\A trj I6n nh^t va glA trj nh6 nhift - Phan Huy KhSi Cty TNHH MTV DWH Khang Vi^t Cong tirng ve (2), ( ) , (4) va c : P < 2x + yjyz P< 1 7= + j-y-J^ 2+ 2y + y/zx X y + 7(y + z)(y+ x) (10) y/x + yfy + : (5) +- z >/7 LSp luan tiTdng tif, la c : 2z + ^fxy z + yl(z + x)(z + y) 2.j-\j-y- (11) \f\+yly+y/z pa'u bling (10), (11) ti/dng tfng xay o y = Vzx ; z = ^ x y Dau b^ng (5) xay o (Jong thdi c dau bang (2), (3), (4) f -5, ; o x = y = z Cong tirng ve'(8), (10), (11) ta c : P < (12) Dau bkng (11) xay dong thcJi c dau bang (8), (10), (11) iL ie uO nT hi Da iH oc 01 / o ,., D a t a = J i - ; b = / - I - ; c = j - J - , thi a > 0, b > 0, c > va abc = ly Luc VP(5) = x = 7yz vy , + a - + + b +2- + c Theo bat d i n g thiJc Cosi thi: ab + be + ca > 3>/(abc)^ = (7) Tir (6), (7) suy ra: VP(5) < (8) P < ^: bo x + V(x + y)(x + z) ^ x f y + z + , ^ ) < Ta s/ ,/z^ - z ^ + x y + Ta c : x ' - x^ + - ( x + 3) = x ' - x ^ - x + = x M x ' ^ - l ) - ( x - 1) = ( x - l ) [ ( x ^ - l ) + (x-^-l) + ( x - l ) ] = ( x - l)^{x^+2x^+3x + 3) (2) Do X > => x ' + 2x^ + x > 0, nen tiir (2) suy ra: x^ - x^ + > x + ce ' ww X \ly^-y^ + y z + = (x-l)[xMx^+x +l ) - ] =(x-l)(x^+x^+x^-3) => x ' - x^ + + x y > 3(xy + X + 1) fa w Ta eo each giai khac nhtf sau: Hudng dan giai Bai toan la stf ket hdp giffa vice suf dung cac bat d i n g thtfc Bunhiacopski, bat dang thiJc Cosi Do la mot bai toan tdng help (1) -x^ +3xy + ' Nhqnxet: P= ok V a y m a x P = x = y = z > , " ; Bai Cho x, y, z la ba so di/dng va xyz = Tim gia tri Idn nhaft cua bieu thiJc: c om Dafu bing (8) xay a = b = c = o X = y = z > up (6) (do abc = 1) ro + 4(a + b + c)(ab + bc + ca) + (ab + bc + ca) Vx (7y + Vz) ^ ; thtfcay! /g + 4(a + b + c)(ab + be + ca) + That vay (8) x^/x + Xyjy + x = y = z>0 /., (11) Tuy nhien no kh6ng tif nhien d cho vi lai xuat phat tijf cac ba't ding I + 4(a + b + c) + 2(ab + bc + ca) + abc Ta se chtfug minh \d\, y, z > 0, thi: o Binh luan: Cach giai chi difa vao cac bat dang thuTc biet triTdc (8), (10), 12 + 4(a + b + c) + (ab + be + ca) ' • y = yfzx Vay max P = o x = y = z > Ta thu lai ket qua tren (2 + a)(2 + b)(2 + c) / t z = 7xy (2 + b)(2 + c) + (2 + c)(2 + a) + (2 + a)(2 + b) ' h ! V- yj\ (8) yfx+yjy+ylz < xVx + Vx•^/(x + y)(x + z) +y ) ( ^ + v VI (9) dung, vay (8) dung Dau b^ng (8) xay x = yjyz J < ^ x - x + + 3xy ^ - • (3) V3(xy + x + l ) Dau bkng ( ) xay o dau b^ng (2) xay 2P = x + y + + ( ^ + y + N ^ ) - , x^ + A p dung bat d i n g thufc Bunhiacopski, ta c6: ^ xy + X (1 + + 1)> , + -+ zx + z +1 ^ ; ^ x y + x + l Vyz + y + N/ZX -+yz + y + +1 ^xy + \2 1 X +1 ' ^yz + y +1 -Jzx + z + l xy + \ v'-, D a u bling (7) xay o +1 + •yz + y + Ta zx + z + l :^:t;vv^ i xy X xy + x + ' + xy + x ' s/ - om xy + x + c , dong thcti c6 da'u bkng (6), (7) ce V a y max P = l o x = y = z = l (3), (4), (5)) va bat dang thiJc w thuTc fa Binh luqn: B a i la sir ket hdp giffa viec suT dung mot bat dang thtfc da bici trirdc (trong bai la cac bat dang (4) ; = Vz (5) ww Bunhiacopski C a i kho c( day la phat hien (3), (4), (5) NhiT vay mot Ian nffa cac ban thay de giai mot bai toan t i m gia t r i Idn nhiil va nho nha't nhieu k h i phai kheo 16o ke't hdp nhieu phU'Ong phap khac nhau, chur khong ddn thuan chi dung mot phiTdng phap la d u ! B a i 10 Cho x > 0, y > 0, z > va thoa man dieu k i e n : x + y + z = x = y = z = phap them bdt hang tuT k h i dung ba't dang thiJc Cosi -) /; Bjki 11 Cho x, y, z la cac so thirc khong a m va thoa m a n dieu k i e n : + y^ + = T i m gid t r i Idn nha't cua bieu thufc: p = X +2y + y'^+2z + z +2x + HUdng ddn gidi + > 2x; y H > 2y; z^ + > 2z, nen ta cd Vi p ^ — ^ — + 2(x + y + l) — I — + 2(y + z + l ) — f — 1,, V- 2(z + x + l ) Da'u b^ng (1) xay r a o x = y = z = l (khi thoa m a n + y^ + z^ = 3) Ta cd: X+y+1 y+Z+1 Z+X+1 T i m gia t r i nho nha't cua bieu thtfc: P = >/x + 7y + Vz - (xy + yz + z x ) = 3- HUdng ddn gidi ^ , (x + y + z ) ^ - ( x ^ + y ^ + z ^ ) 9-x^-y^-z^ Ta c6: xy + yz + zx = ^ '- = , Nhqn xet: Mau chot cua bai toan la du'a P ve dang (1), sau suf dung phiTdng bo x = y = z = l z^ + %/z + >/z > 3z v a y m i n P = 0x = y = z = l (10) ok Da'u b^ng (10) xay o y x + l + xy ^ + x + xy _^ Thay (9) v a o ( ) va c6: P < (3) Tilf(l), ( ) c : P > up + • + ^^^y X = y = z = (chii y luc thoa man: x + y + z = 3) o (8) ro yz + y + (2) Da'u bkng (5) xay /g -+ xy + x + 1 3x y, z > 0) X, Dau bang (6) xay o > x^ + y^ + z^+ 2(Vx+7y+%/z)>3(x + y + z) = = y = z = (do xyz = 1) i + ^ + ^ xy + x + yz + y + I zx + z + i + N/X () Cpng turng ve (1), (2), (3) ta c6: (7) 1; y = 1; z = (do X = zx + z + l) V i x y z = l,nen: y' + ^ (1) Da'u bang (2), (3), (4) tiTdng iJng xay k h i x^ = Vx ; y^ = xy + x + = yz + y + = zx + zx +1 X Tiif (6), (7), ta c6 P < X + Z + A p dung bat d i n g thiJc Cosi, ta c6 iL ie uO nT hi Da iH oc 01 / N/X ) T f i f d ) , (2) s u y r a : y +1 x+y +1 y+1 ^x+y+1 z +1 y+ Z + + 1- z+1 x+1 y+z+1 z+x+1 x+1 z+x+ U CtyTNHH MTV DWH Khang Vi^t Chuy6n BDHSG To^n gia trj lOn nhft va gia tri nh6 nh^t - Phan Huy Khii p 2 Ttf (2) dung suy (1) diing Nhan xet dtfdc chiJng minh R6 rang ba so x + y^, y + z^, z + x^ c6 it nha't hai so hang cCing dau Vi thd' c6 the gia sur (ma khong lam ma't tinh tong quat) (x + y^ )(y + ) > Ttf (2) v£l theo (1), ta c6: ^ l + x + y^ + -y/l + y + z^ ^ + ^ l + x + y^ +y + z^ TCf suy ra: Jl + x + y^ +-y/l + y + z^ + Vl + z + x^ > l + l + x + y^ + y + z^ +>/l + z + x^ => P > >/I + ( N / I - Z + Z2)77 yjiyfU^f+x^ ^ ^' (3) (Chu y: X + y = - z V I x + y + z = 0) Ap dung nhan xet sau day: Vdi moi so thi/c a, b, c, d ta c6: V a ^ + b ^ + V c ^ + d ^ >yl(a + cf +(h + df > i (*) Tiydo ta c6: ^ y+1 z+1 x+1^ x + y + y + z + z + x + 1^ (x + l)2 2L(y + l)(x + y + l) (z + l)(y + z + l) (x + l)(z + x + l) Theo bat ding thtfc Svac-xd, ta c6: i , ; , ^ ^ (y + l)2 , (z + lf (x + lf ' -+• (y + l)(x + y + l) (z + l)(y + z + l) (x + l)(z + x + l) (y + l + z + l + x + 1)^ (4) (y + l)(x + y +1) + (z + l)(y + z +1) + (x + l)(z + x +1)', De y r^ng do: x^ + y^ + z^ = 3, nen ta c6: (y + l)(x + y + 1) + (z + l)(y + z + 1) + (x + l)(z + x + 1) t '"^ , = 3(x + y + z) + xy + yz + zx + x^ + y^ + z^ + J, = - (x^ + y^ + z^ + + 6x + 6y + 6z + 2xy + 2yz + 2zx) = - (x + y + z + 3)^(5) , (z + lf iL ie uO nT hi Da iH oc 01 / (y + lf + •p'-i it i Ta i > (x + l)(z + x + l) bo ok c (y + l)(x + y + l) (z + l)(y + z + l) Da'u b^ng (6) xay o X = y = z = Tir(3), (5)suyra:P < ^ yWfiS Dau b^ng (7) xay o dong thdi c6 da'u b^ng (3), (6) X = y = z = up + ro + Vay max P = •^ 0; a, b v^ a + b deu > -1 Khi d6 Vl + a + Vl + b > l + Vl + a + b That vay sau binh phiTdng ca hai ve cua (1), ta c6: (1) 2 + a + b + 2V(l + a)(l + b ) > + a + b + 2Vl+a + b (!+a)(l+b)> + a + b o a b > s/ (z + 1)^ /g Tir(4), (5)suyra: (y + 1)^ om TV/.t^ (2) +,J{J^f + J^2 > ^ ( i _ z + z2 + Vf+z f + (x + y)^ (4) Tir(3), ( ) v a d o x + y + z = 0,nenc6: P > + v ( V l ^ z + z^ + VT+z) +z^ (5) Talaico: ( V l - z + z^+VrTz) +z^ « = l - z + z^ + l + z + V ( l - z + z^)(l + z) + z^ = + z ^ + V l - z ^ + z ^ + z ^ =2z2+2 + 2Vl ^» •< -jjj^iS + z^ (6) Ta se chtfng minh: 2z^ + + 2Vl + z^ > '" That vay (7) o z^ +1 + Vl + z^ ^ o Vl + z^ > - z ^ " Do - z^ > nen (7) » + z' > - 2z^ + z" o + 2z^ - z* ^ (7) o z K z + 2-z2)>0 (8) Do z e [-1; 1] => z - z^ ^ -2 z + - z^ > 0, vay (8) dung => (7) dung Tilf (5), (7) suy P > (9) Da'u b^ng (9) xay de tha'y o x = y = z = (cac ban tif nghi$m lai) Vay minP = 3«x = y = z = ^inh luan: Bki toan thUc stf la bai todn tong hdp Ma'u chot Ik dxia vao bat ding thtfc (1), de suy danh gid (3) Lai drfa vao ba't ding thuTc hien nhien (*), de It c6 danh gid (5) Sau lai difa v^o bat ding thuTc (7) de c6 danh gia (9) ^^i 13 Cho X, y, z Ik cde so thiTc diTcfng va thoa man dieu kidn: xyz = Tim gid tri nh«5 nha't cua bieu thtfc: iit ' ' V ,55 ChuySn dg BDHSG ToAn giA tri Idn nhSt va gi^ trj nh6 nhaTt - Phan Huy KhSi Cty TMHH MTV DWH Khang Vi$t pai 14 Cho x, y, z, t la cac so diTdng va thoa man dieu kien: xyzt = V ( i + x-^)(i + y ' ) V{i + y )(i + ^ Tim gia tri nho nhat cua bieu thiJc: z ) ( i + x-^) HUdngdangidi v i > ^~ Ap dung (1) ta c6: (1 + af ^ 4z' 4z2 ""77:: i (1) 7:;- ; Y l + 2b + b^+2a + 4ab + a^+2ab(a + b) + a^b^ l + ab(a2+b^)>2ab + a V , (2) liiv : (3) Dau b^ng (3) xay o a = b s/ (2 + 4a)(2 + 4b) (2 + 4b)(2 + 4c) (2 + 4c)(2 + 4a) c ( i + 2b)(l + 2c) ( l + 2c)(l + 2a) Vay 2a^b^ + > 2ab + a ' b l l + 2(a + b + c) + 4(ab + bc + ca) + 8abc Dau bkng (4) xay o ab = *t " Da'u bkng (1) xay o dong thdi c6 dau bang (3), (4) , j ^ i ' ' *' Da'u b^ng (1) xay ww w Ta c6: a + b + c > 3^/abc = 3, - Tir (2), (3), (4) suy (2) dung, ttfc Ih (1) dung + xy fa ce • (4) , Ap dung nhan xet (1), ta CO: P > — + bo a + b + c + 2(ab + bc + ca) it' • o a = b = ok ( l + 2a)(l + b ) ( l + 2c) ab + be + ca > 3^(abc)^ = 3, X6t hieu: (2a^b2 +1) -(2ab + a^b^) = a^b^ - a b +1 = (ab - f c a(l + 2c) + b ( l + 2a) + c ( l + 2b) /g ( l + 2a)(l + 2b) b om a up 4c ro 16b ^'p^-^ i = ^ l + ab(a^+b2)> a V + l abc = 16a P>4 i i D o a > , b > n e n a ^ + b^>2ab Luc tiif (3) suy hay P > 1_ (1 + b)2 " + ab o Khi do X, y, z > va xyz = => a > 0, b > 0, c > P> > < z^ 1 , + 2(a + b) + a^ +b^ +2ab + 2ab(a + b) + ab(a^ + b ^ ) > " ( + x2)(2 + y ^ ) ^ ( + y2)(2 + z ) ^ f z ) ( + x ) " x^ « Ta ' , 4y2 ^ T h a t v a y : ( l ) o (2 + 2a + 2b + a + b ) ( l + a b ) > ( l + 2a + a2)(l + 2b + b2) ' V4(l + x-^)4(l + y ^ ) ^ V ( i + y ' ) ( l + z ' ) ^ > / ( l + z ) ( l + x ) 4x^ HUdngddngidi iL ie uO nT hi Da iH oc 01 / 4y^ f - Ta CO nhan xet sau: V d i moi so dating a, b ta co: Do (2) dung nen (1) dung ^' l.t ; That vay (1) o + a ' < a'* + 4a^ + o a" - 4a'+ 4a^ > o a\ - 2)^ > , 4x^ (1 + x ) ' ^ ( + y ) ' " " ( I + Z ) ' ' ' ( l + t)2 • N M n x6t: Va thi ( l + a"*) a + b + c + 2(ab + be + ca)> = + Babe ^ 1 Tac6: x =y=l z=t=1 + zt + l + xy + + xy => (1 + Babe) + [2(a + b + c) + 4(ab + be + ca)] o = + zt ( l + xy)(l + zt) —!— ' " - V x = y = z = t = l ? i + xy + zt , = (do xyzl = 1) + xy + zt + xyzt Vay tir(4)e6: P > | thay dau b^ng (6) xay a = b = e = l < = > x = y = z = V^y P = — C:>x = y = z = " (5) (4) '\.,.X + zt (3) i gfi i ' ' Dau bang (5) xay x = y = z = t = Vay P = l o x = y = z = t = l ^hgn xet: Quan nhat Ih biet diing bat d i n g thiJc phu (1) fi^i 15 Cho X, y, z la ede so' thifc diTdng va thoa man dieu kien: x y z x + y+ z > - + —+ — y \ ' ; f > • 157 Cty TNHH MTV DVVH Khang Vigt Chuy6n BDHSG Toan gia trj Idn nhift \ik g\i trj nh6 nha't - Phan Huy Khii Tim gia tri nho nha't cua bieu thiJc: P = + x +1 + y + 3(xy + yz + zx) + 3(x + y + z) = 2(xy + yz + zx) + (xy + yz + zx) + 3(x + y + z) z+1 Do (5) suy ra: 3(xy + yz + zx) + 3(x + y + z) > 2(xy + yz + zx) + 4(x + y + z) Thay (10), (11) v^o (9) suy VP(2) < - , tCr d6 theo (2) c6: • r>i th Hiidng dan giai V i e t l a i P d i r d i dang: ^ , Taco: +l y ^x + 1- + y z+1 (y + l)(z +1) + 1 + + x+1 • y+1 •z+1 (X + ! - + z- + 1^ + l)(z +1) + (x + l)(y + l)(z + l ) , { , V y z xy X yz zx ,x J • >, i / , DSu bkng (13) xdy (2) O dong thcJi c6 dau bang cac bat ding thtfc (5), (6), (7), (8), (11) z o x =y= z=l, (3) Turgia thiettaco: x + y + z > - + - + y z x X y z x^ y^ z^ (x + y + z)^ nhiTng: - + i + - = — + 1- + — >1 i _ z+1 T t f ( l ) , (12) ta c6:P > - xyz + (xy + yz + zx) + (x + y + z) + l X y+1 Dau bang (12) xay rax = y = z = l + l)(y +1) (X (xy + yz + zx) + 2(x + y + z) + ' ', x+1 (1) iL ie uO nT hi Da iH oc 01 / P= 1-x+1 Vay P = - o x = y = z = l (4) Binh ludn: Ro rang day la bai todn long hdp, sijr dung rS't nhieu bat d i n g thuTc xy + yz + zx Ta phu, bat dang thuTc Cosi, bat ding thiJc Svac-xd, d^ gidi bai loan (theo baft d i n g thtfc Svac-xcJ) => xy + yz + zx > X + y + z (x + y + z) X+ y +z -=>1> xy + yz + zx xy + yz + zx s/ + y + z> up X ro Tir (3), (4) suy ra: • X 1 -^z X y x y z 2- + - + - 2- + + y + z > - + i - + - = _ - + - +— 3l Z X; 3k X y) z> y z X 3l y ^ = y y ^>3 ce y X — = • , nen tUOng i\i ta c6: yz ^xyz 2^ + i ww ^/xyz '4xyz Tir (6), (7) suy ra: x + y + z > ^ + y + z _^ ^/xyz ^^ D^u b^ng (8) xay r a o x = y = z = l Tac6: VP(2) = +V y + ( z •+ + x)^ ' 'yz''+(x + y)^ ' Hiidngddngidi ' ^' ' Ta c6 nhan xet sau: vdi moi x, y, z la cac s6'thifc dufOng, ta c : ' ' ' V x ^ + ( y + z)"' Thatvay:(1) x^+y^+z^ » x ^ + ( y + z)^ (x^+y^+z^) w fa VI: X bo ok Ta lai co: lx^+(y + z)-' c om Dau bing (5) xay r a o x = y = z = l r,, P= /g (5) Bai 16 Cho x, y, z la cdc so thiTc diTdng Tim gia tri nho nha't cua bieu thiJc: (xy + yz + zx) + 2(x + y + z) + o x^ [x^ + 2x2 (y2 + ) + (y^ + z^) J > x U x ^ x + z)^ o x + ( y + z ) + (y2+z2) >x(y + z)^ ai? - - Theo bat d i n g thtfc Co Si, ta c6: 2x2 + x H y ' + z ' ) + ( y ' + z2 f > 2^/2x2 ( y ^ + z ^ f xyz + (xy + yz + zx) + (x + y + z) + _ 2(xy + yz + zx) + 4(x + y + z) + ' 3xyz + 3(xy + yz + zx) + 3(x + y + z) + Tilf (8) suy ra: 3xyz + > 6, Ro rang: 2(y2 + z^) > (y + z)^ ?, Tir (3), (4) suy ra: 2x2(y2+/.-) + (y2+z2) > Jx2(y + z)^ = x ( y + z)^ Tir (4) suy (2) dung, vay (1) dung Da'u bang (1) xay (3 - 2c)(3 - 2a)(3 - 2b) < abc : (6) •" ' =^ 27 - 18(a + b + c) + 12(ab + be + ca) - 8abe < abc => 12(ab + bc + c a ) - a b c < 4(ab + be + ca) - 3abc < Dau bang (6) xay a = b = e Ta co: ab + be + ea < (a + b + e)^ Dau b^ng (6) xay o (6) -iv::::',.: = (7) a = b = e Tur (6), (7) co: 5(ab + be + ca) - 3abc < 12 (8) Tir (4), (5), (8) suy ra: 4P < 12 =:> P < V* o a V a y max P = o ' ' dong thcJi co da'u b^ng (6), (7), (8) = b = c= X TCr (2), (3) suy ra: P > - V a y m i n P = - Gia trj nho nha't dat diTdc k h i x = y = ^ ; z = izUxUl) o Da'u bkng (9) xay o Tird6c6:Q < - |y 4P < x^-* (x^ + y-* + ) + y^z"* (y'* + y' + ) + y.\' - Ta J :^^ 'K^\ J.t^l^ s/ + up I ro + zV(z-' + x ^ + ) > z V Cong tirng ve (1), ( ) , (3) va c6: T i m gia tri nho nha't cua bieu thii'c: (2) Da'u bang ( ) , (3) tu'dng uTng xay o B a i 17 Cho x, y, z > va ihoa man dicu k i c n x + y + z = p= TiTdng tiT, ta c6: x''z''(y'* + /* +2)> 4y'*z\ = = (9) Cty T N H H M T V D W H ChuySn 6i B D H S G Todn gia trj I6n nh9't vS gia tr| nh6 nhjl - Phan Huy Kh^i Nhdnxet: , >.,() ^4 •( y'i: abc > 0) (vi y ' Dau bang (5) x a y Chi C O hai kha nang: Vay max P = a) Co hai Ihiifa so am, mot ihuTa so dufdng Gia suf ^ ^ =2 'Hfitr'!, = 2; ' y = 1.- { ic) ,;,|.: :f (6) b < = : > b < ( v i V I bSO) X = 1; y = _x Khang V i j t •'-'OJ ^ ^ Do vai tro bmh dang giffa x y, z nen ta c6 the gia suT y la so hang giffa < t' [b + c - a < => khong xay irtfdng hdp n i y ( b) Cabathirasodifting ^ / v > r-, H^' < ".mMxy %tc ba so X, y z Khi ta c6: Q_x ^y_^z_^j^x ^z_ Dat X = a + b - c; Y = b + c - a; Z = c + a - b =^ X > 0; Y > 0; Z > y X ~ > i S!)x,.~H That vay (7) Ta c6 theo ba'l ding thiJc Cosi: (X + Y)(Y + Z)(Z + X) > 8XYZ =>8abc>8(a + b - c ) ( b + c - a ) ( c + a - b ) z ' - * "t - ' ! Mi',m^^:Mi 'Adi'h^ iA x ,V; ' •• •> sftVidflu! XJ^^^w*!' • ' : v'- ' oiiy,i-^>()^^y-y'^y^-'-^>o^o / y yz yz => (6) diTcJc chuTng minh! Bai 19 (8) Do y la so hang giOfa nen (8) dung vay (7) diing X, y V Z y z X s/ X om /g ro up Cho X, y, z 11; 2] Tim gia tri Idn nhat cua bieu thiJc P = - + i - + - T i r ( l ) s u y ra: < ^ vy/ + 1 \ ok w ^ X ww X bo ce V i = i < » x = l;y = c n - = o ly y Cong lifng ve(2), (3) va c6: X y —+ — ly X, si M> • iiA/-? ^r/ih jf|>r;x ,r,Jr _x = - ; y = - l = v v « , T\m gia Iri nho nha't cua bicu thiJc: P = x(y + /.) + /(x + y) V i c t lai P dvtiVi diing: P = (xy + yz + zx) + xz (1) Dat P| = xy + yz + zx; P = xz K h i lir ( I ) c6: P = P, + Pj (2) Ta => + 2P| > ( ) x^+y^+z^^] /g om c bo ww Do he N 1 (6) Q'j (2) 2(x + y + z) y- z xy (3) yz zx Da'u bang (3) xay x = y = z = ' Tir(2)(3)suy r a P > \ xy + •+ yz zx (4) 2(x + y + z ) Da'u bang (4) xay dong thdi c6 da'u bang (2), 6) X = y = z = Do xyz = 1, nen tif (4) rut gon ta c6 P > y=() X = X O fa +y +z = I (x + y) + ( y + z) + (z + x ) X + y+ z w x-+y^+z^ = l (5) ce TCf (4) suy ra: m i n P = - ^ < = > y = () D i 1 1 Ro rang ta C O — + — + — > — + — + — (4) =:> x ' + y ' + z ' + 2(xy + yz + zx) > x^+y^+z^ =l I T h e o bat dang thufc S vac-set, la c6 P > HU('/Hg dan gidi TiT (2) suy ra: m i n P| = - — HUi'fng ddn gidi iL ie uO nT hi Da iH oc 01 / p; Bai Cho X , y, / la ba so thifc thoa man: X ' + y ' + P, z"*(z + x ) ,, X , y cung dau h • y"*(y + z) x| = ; y | = l Vay max P = + —• i t L a i theo ba't dang thilTc Cosi, ihi x + y + z > 3^/xyz = (5) (6) Dau bang (6) xay o x = y = z = -z C O nghicm (chiing han x = — ; y = 0; z = — ^ la nghicm cua he (6)) Tir(5)(6)c6 P > ^ Da'u bang (7) xay o dong ihc^Ji c6 da'u bang (5) (6) Vay theo tinh chat cua gia t n \(1n nha't va nho nha't ( X c m bai - chtfc/nj; O X =y =z=1 cuon sach nay), la c6: Vay minP = ^ < = > x = y = z = l P = P| + P: = - - - - = - 2 (7) Chuygn 6i BDHSG ToAn gi& trj Idn nhJt va gii tr| nh6 nhat - Phan Huy Khii Nhdn Cty TNHH MTV DWH Khang ViQt xet: P= Day la bai loan long hdp, suf dung den nhieu ba't dang thtfc S v a c - x d , Cosi,., qua trinh giai toan Ta C O each giai khac nhiTsau: i/rt»|v.j x * < 7x-^ + 2y^ + Y^'YZ Z ^ Z X Y +Z + Z+ X (doXYZ=l) = X'* (X + Y)Z -.4 ,.:i:::l :r ?.l-Ji:*;i: T h c o ba't diing thi?c Cosi, ta c6 Y^ + b| = (Y + Z)X + • (X) (Z+X)Y b2 (9) < Ttr.ngt,,c (10) (Z+X)Y^^2 : • ^x''+2y^+6 y-' + z ' + ^x-'+2y'+6 n i ^ Vz-V2x-V6 j , o„3 z'+2x'+6 1 y^+2z^+6 ' z^+2x^+6j (1) irj irs Ta , 7y"^+2z-V6 hayP< J3 z^'' Vz-%2x-^+6 J\^+2y^+6 ^ > X^ =• = b j = 1, ta c6: rx+ YIZ +^ giai 7x-^ + 2y^ + ' ^ V y ^ + z - ^ + ' iL ie uO nT hi Da iH oc 01 / X +Y + A p dung bat d i n g thiJc Bunhiacopski v d i hai day X Y Z = l ( d o x y z = 1) X-^XY yjz^+2x^+6 HUdng dan e a t X = — ; y = — ; z = — k h i ta c6 X , Y , Z la cac so thiTc du-cfng va X ^ Y Z > ) I lie P = yjy^ +!•/?+6 + V"*^ + Z2 ' ' • " up + ' ' (12) /g p + XY + YZ + ZX > ' ~' ro Cong tirng vc (9) (10) (11) va c6: s/ Da'u bling (1) xay x = y = z ,^ c C:>X = Y = Z = om Da'u bang (12) xay o dong thdi c6 dau bang (9) (10) (11) Do X^ + Y^ + Z ' > X Y + Y Z + Z X , ncn ta c6 tir (12): ok ' fa Dau bkng (13) xay r a c : > X = Y = Z = w L a i thco bat dang thtfc Cosi thi X Y + Y Z + Z X > ^ ( X Y Z ) ^ = " • (14) ww « - x- " ^ -SlMilr; (4) + (15) + yz + z + i (5) zx + x + Da'u bang (5) xay o dong thdi c6 da'u b^ng (1) (4) x = y = z = l Do xyz = 1, nen de tha'y 1 •+ +• xy + y + yz + z + zx + x + Tir (5) (6) suy P < V a y minP = ^ x = y = z = Ta thu l a i ket qua tren V a y m a x P = l o x =y=z=i T i m gia trj Idn nhat cua bieu thuTc (2) (3) ••-.•I'M;) '••! ! ) f^H ! Thay(2)(3)(4)vao(l),vac6P< j Da'u bkng (15) xay o dong thdi c6 da'u bkng (13) (14) i , X = Y = Z = l o x = y = z = l B a l 23, Cho x, y, z la cac so thi/c diTdng thoa man dieu k i p n xyz = \in| • lUi^HiU Da'u b^ng (2) (3) (4) ti/cfng uTng xay r a o x = y = l ; y = z = l ; z = x = l ; , r w ! gn/iH;??-' (13) ^ = > x S y ' + 6>3(xy +y+1) z^ + 2x^ + > ( z x + x + 1) bo + YZ + ZX •''^ x ' + y ' + = ( x ' + y^ + 1) + ( y ' + + 1) + > 3xy + 3y + ce XY '1'' ' ' ' ' A p dung ba't dang thtfc Cosi, ta C O xy xy + y + -+ ^ = (6) xy + y + xy + y + -• , Dau b^ng (1) xay o x = y = z = 1, • •^ , t (7) ^ ' ^- Cach giai tren la sir phoi hdp giffa viec suf dung ba't d i n g thiJc Bunhiacopski, ba't d i n g thiJc Cosi va bien ddi dai so ddn gian 167 Chuyen dg BDHSG loin gJA tri I6n nha't Cty TNHH MTV DVVH Khang M i ^ gia trj nh6 nha't - Phan Huy Khi\ plnh luan: Cach giai c6 dicu khong lu* nhicn c'J cho: lam la lai b i c i c6 X c t each giiii khac sau day: Thco bat diing thtjTc Cosi ta c6: + danhgia (14) x-+2y-+f) ^x' + 2y'+6 " t I I +—+2 +2 >9 + y• +2 x'^(y + z) y ^ • ^ + l/.yfy y"(z + x) ^ Z\/Z+2XN/X , ^ , l^jy/ = ^ + (9) y +2 Ttf T i r ( ) (9) suy sjx^ + 2y^ + x-^ + y + V > Da'u bang (10) xay o dong thcti c6 da'u bang (8) (9) o Tifdng tiT, ta C O 7y^ + 2/;^ + (15) Vay (15) diing o y ( ' ^ + x) > y.\f/.+2\\lx (16) 2y^/y ^ 2ZN^ y7y+2z%/z ^ y^y ' ^ ZN/Z+2X\/X (3) x7x+2y7y ' xVx (2) Z\/Z+2XN/X Cong tifng vc (1) (2) (3) va c6: 3x'yV.- = 2XN/X Dau bang (4) xay dong thdi c6 da'u bang (1) (2) (3) (y* + 2)(z' + 2) + ( x ' + 2)(z' + 2) + ( x ' + 2)(y' + 2) < ( x ' + ) ( y ' + 2)(z' + 2) + z'x' + 4(x' + X \ / - = XN/x+2y7y (14) o yV,' ^ > z ^ x + y) P> bo Ta sc chiirng minh —-^— + —r^— + — < x-%2 y-+2 z/+2 z) up ro /g om , +— +1 + z-^ + p That vay (14) c X- +— ok + z) + Lap Imin tiTdng lif, C O (12) tirdng i^ng c6 y = /;/ = \ Cong tifng vc (10) (11) (12) c6 6P < x'(y (do xyz = I ) Da'u bang ( I ) xay y = z (1 +— + 1, z-^ + Da'u bSng (11) (12) xay o y7y+2z7z +2 suy x^(y (10) Ta x-%2 s/ • < ^ x-%2yU6 iL ie uO nT hi Da iH oc 01 / Hitiing Mil gidi |P ' ' X^ Y^ XY+2ZX YZ+2XY Z + 2X X + 2Y -72 ^ (5) XZ+2YZ " (5) va thco ba't dang thu-c Svac-xd, ta c6 P > 2- (X + Y + Z ) ' (6) 3(XY + Y Z + ZX) Da'u bang (16) xay x = y = z = ^a'u biing (6) xay r a o X = Y = Z = l < » x = y = z = l TCr ta C O maxP = x = y = z = Ta thu lai ket qua Ircn '^orang(X + Y + Z ) ' > ( X Y + Y Z + ZX) O) 169 ChuySn dg BDHSG Join gJA trj I6n nhjft va gii tr| nh6 nhtft - Phan Huy KhAi Cty TNHH MTV D W H Khang Vigt D a u bang (7) xay raX = Y = Z = l xy^ ft'; Tir(6)(7)c6P>3vaP = o X = Y = Z=l V a y minP = c > x = y = z = l " " ^ ^ • ' — + >/y^ + x^+2 v,,:r-i- ->y2 + > | y ^ , :SiimiPA(Udt:> z ^ + > - z ^ , (1) D a u b^ng (1) xay y + = y^ - y + Tir(7)(8)(9)(10)suyra -" < y" - 2y = _^ /g > 2y z 2z V^y.yz.yz s (4) bo l,y'+2 fa P> w Cpng tirng ve (2) (3) (4) va c6 y X z^+2 z x'+2; (5) = y = z = - , Ta se chiJng m m h That vay (6) (do X 2y 2z + ^ +— >2 y^+2 ' z^+2 ' x^+2 X - I + y + z = 6) 2x y^ + y- 2y z2+2 ^j.,^ ^ 'i'!^)' => xy + yz + zx < 12 (6) (13) I x2+2 y ' + (9) V T (7) ^ ^ ^ + , r ^ ^ + ^""^ -y^/4y -z\/4z -x\/4z ^ 2 hay V T (7) < ^ ( ^ x x y x y + ^ly.yz.yz (doy>0) 2x X ra(dox>0) (8) ••• n*W DSu b^ng (8) (9) (10) Wdng tfng xay o Tir(l)suy ^.w^ V '^"^^^ iL ie uO nT hi Da iH oc 01 / TiTdngtiTco x ^ + > - x ^ , L^xin Vx-^ + A p dung ba'l d i n g IhuTc Cosi, la = dung ba'l d i n g Ihtfc Cosi, la c6 2y^ + = y^ + y^ + > ^ / / = y ^ Hudng dan gidi o y ,, „, ' — + sl/.^ + zx^ y2+2 - ' • ' ' B a i 25 Cho x, y, z la ba so thiTc diTdng va thoa man dieu k i e n x + y + z = T i m gia t r i nho nhat cua bieu thuTc P = yz^ x = y = z = Do (7) dung nen (6) dung va dau b^ng (6) xay o Tir (5) va (6)^c6 P > x = y = z = (14) Dau b^ng trohg (14) xiy dong thdi c6 dS'u b i n g (5) va (6) x = y = z = T o m l a i minP = o x = y = z = ni Cty TNHH MTV DVVH Khang Vi§t Chuyfin dg BDHSG Toan gii tr| lOn nhat va gii trj nh6 nha't - Phan Huy Khjii Nhdn xet: Ro rang day la bai loan long hilp suT dung iCr ba'l dSng ihiifc Cosi, dc,i dUa vao mcH bat dang ihiyc b i c l lrU'('Jc (6) B a i 26 Cho x, y, / la ba so diTdng va thoa man d i c u k i c n — + — + - < ! X y /, 7^y" + + 14yz , ' N/3Z^ + Sx" + 14/x x = y = z = j j^i^^j T i r ( ) (7) la C O P ^ - • iL ie uO nT hi Da iH oc 01 / x - +8y^ + i x y ' Dau bang irong (8) xay dong thcJi c6 da'u bang (6) (7) • '^ ' X Ta C O ihco ba'l dang ihuTc Cosi •• ^^'^^ " V x + 8x^ y-+14xy 2xx^ + 3y NhiTvaysuyra , > — Da'u bang Irong (1) xay o x + 4y = 3x + 2y o x = y 2 TiTdng lir, la c6 , ^ > , V3y2+8/.'+14y/, 2y + 3z (4) bo ok —+ -^—- +—• 2x + 3y 2y + 3z 2z + 3x ce C:> X = y = z 2x + 3y ^ _ y L _ + _i^l_> 2y + 3z 2z + 3x Tir (4) (5) suy P > ^ (X Ncu a > 0, b > thi ro rang + y + z) Da'u b^ng (6) xay o dong thcJi c6 da'u bhng (4) (5) hay a b" ;" 1 (a + b)- > — a b ab (a + b ) ' > (1) Dau bang irong (1) xay o a = b Do vai tro binh dang giiTa x, y, z ncn c6 the gia suT x > y > z L(x-y)' (2x + 3y + 2y + 3z + 2z + 3x) x^ 7^ = > — +— +— > - ( x + y + z) 2x + 3y 2y + 3z 2z + 3x Da'u b^ng (5) xay x = y = z ' Ta C O nhan xet sau day: (x + y + z)^ ww x^ !—- + ( y - z r ( z - x r A p dung (1) cho cSp so x - y > 0, y - z > 0, ta c6 w Theo bat d^ng ihuTc S v a c - x d , la c6 (x-yr HU(?ng ddn gidi om c z^ fa , s/ /g (3) ! — + Ta T i m gia tri nho nha't c u a b i c u thuTc P = D a u bang irong (4) xay o dong Ihdi c6 dau b^ng (1) (2) (3) i-; V „ Bai 27 Cho x, y, z la ba so thiTc doi mot k h a c trcn d o a n |(); 2| (2) D a u bllng (2) (3) liTttng iJng xay y = z; z = x Congtirngve'(l)(2)(3)vac6P> thiirc S v a t - x ( l (1) > - ^ — N/3// + x ' +16zx 2/ + 3x x^ Shan xet: B a i l o a n la kc't hdp c i i a v i c e sii" dung bat d a n g thu-c Cosi, ba't dang ^ 2x 3y up " = y = z = Vay minP = - x = y = z = ro 73x^ 8y^ 14xy ^ 7(x 4y)(3x 2y) > p o O < — + — + - < 1, ncn C O x + y + z > x y z Da'u bang (7) xay T i m gia Iri nho nha'l ciia bieu ihuTc I I Theo ba'l di^ng ihiJc Cosi cd ban, ta c6 (x + y + z) — + — h — X y z -+ - (y-z)^ (5) (x-yr \ [ ( x - y ) + (y-z)r >8 • 1; (2) ( y - z r ( x - z )2 • I^au bang (2) xay r a < = > x - y = y - z (6) ( TiJf (2) suy (x-y)^ hay p > (x-z) • (y-zf (z-xf (x-zf (z-xf (3) Chuyen 6i BDHSG Join gJA tri I6n nhat Cty TNHH MTV DVVH Khang Vi$t glA Iri nh6 nhSt - Phan Huy Khii Dau bang (3) xay « x - y = y - z Do x > y va X , z e [0; 2] =^ < x - z < Dau bang (4) xay o x - z = o x = 2; z = (5) That vay (8) o 272N/XVT+x + y > z t CO P = x^ + y^ + z^ + xyz = (x + y + z)^'+ xyz - 2(xy + yz + zx) i = + x y ( z - ) - z ( y + x) f = + xy(z - 2) - 2z(3 - z) ^ I Hien nhien ta c6 xy < ^ 2x ^x + y ^ ' '3-z ' , (1) \ I Do < z < => z - < 0, vay tijf (1) CO +x va yz < 1 M-J Ta •b nj isJs • Do vai tr6 binh dang giufa x y z nen co the gia suT x > y > z Do < yz < t: (Ban doc tU" nghiem lay dicu nay) Hiidng ddn gidi + y^+z^ De thay da'u b;lng (10) xay r a o x = y = z = ViT^ Hien nhien ta co (9) Vi (9) dung, nen (8) dung Tir do tinh binh dang cua x, y, z ta c6 minP = - ^ l + y ' " l + z2 ' o (V2x->/rTx)^ > 2; z = Ta lai co (8) + 3x - 2V2x(l + x) > 2; y = 1; z = iL ie uO nT hi Da iH oc 01 / X = X = (7) i V f )• Tiif(3) ( ) c P > - Vdi gia Ihiet x > y > z ihi dau bang (5) xay x-y=y-z Cong tCrng vc (5) (6) va co P < / - ^ + • V l + x 1+x ^|2 T a sc chu'ng minh 2., ^ ^' + X + x V2 (4) !(4) P > + (z - 2) r3-z^' I 2J -2z(3-z) "S'u b^ng (2) xay x = y 3-z (z-2)—-4z Ta C O VP (2) = + Mat khac, ta CO X > 0: (6) (2) = 9+^[(z-2)(3-z)'8z]