Tài liệu Chuyên đề bồi dưỡng học sinh giỏi-giá trị lớn nhất, giá trị nhỏ nhất phan huy khải (phần 4)

Cty TNHH MTV DWH Khang Vijl Chuy§n dg BDHSG Toin g\i trj I6n nha't va g\& trj nh6 nha't - Phan Huy KhSi = 9 + ^^[-z' -3 4 • , ,, , TiJf do suy ra P < 2. z-6] Z =X = i ( z ^ - 3 z + 18) = i [ ( z 2 - 3 z + 2) + 16] z=y = i [ ( z - l ) ' ( z + 2) + 16]. 4 Do 0 < z < 1, nen tir (3) suy ra VP (2) > 4 => P > 4. X - y- Z- 1.,.', Sau do trao doi vai tro giffa cac bie'n x, y, z la c6: '^x = y = z = l ,j;>j,j ,ij.,|yrj',{;iy z= ro V i the xet hai triTdng hdp sau: ce y =l fa w. ww Dau b^ng trong (1) xay ra o o X =y=z=1 x^-z2 = 2. Neu X > z > y > 0. Luc do ta c6 y(z - x)(z - y) < 0 o xy' + yz' + zx^ < zx' + zy' + xyz o xy^ + yz' + zx^ - xyz < z(x^ + y^) o P < z(3 - z^) o P < - z ' + 3z - 2 + 2 o P < - ( z - l ) ' ( z + 2) + 2 (1) Tff gia thiet x + y + z + xyz = 4 va ket hcJp vdi (1) suy ra 3z + 7} < 4 < 3x + x^ Tir z' + 3z - 4 < 0 o (z - 1 )(z^ + z + 4) < 0 o z < I . .. • Tirdngtirx^ + 3 x - 4 > 0 o x > 1. (.< Vay ta CO X > , >, ^,.„ , . 1 va z < 1. , , 1. N e u x > y > 1 > z Tff bo l ) % + 2) + 2. rr x - y l_z-y om •/). ok o xy^ + yz^ + z x ' - xyz < y(x^ + Do x^ + y^ + z^ = 3, nen c6 P < y(3 - y') Do vai tro binh dang giffa x, y, z, nen c6 the gia suT x > y > z. D o x > y > z , nen CO hai kha nangxay ra: .c o x^y + xyz > xy^ + zx^ o xy^ + yz^ + zx^ < x'y + yz^ + xyz /g > y > z > 0. Tif do ta c6: x(y - x)(y - z) < 0 Tilfdo suyra P < 2 Ta s/ up vong quanh, nen chi c6 Ihc gia suf x = max{x, y, / ). hay P < - y ' + 3y - 2 + 2 h a y P < - ( y - z = 0. Hudng ddn giai Hi/(htg ddn giai 6 day giu'a cac bien x, y, z khong c6 linh binh di^ng nhufng c6 vai tro hoan vi • ^ •^i-U%y^; •- Bi^i 31. Cho X, y, z la ba so" thiTc duTdng sao cho x + y + z + xyz = 4. Bai 30. Cho x, y, z la cac so ihi/c khong am va thoa man dicii kicn x^ + y^ + z' = 3. 1. Neu 0 x = 0;y = l;z = V2 _x = 1; y = vdi cac phep bien ddi dai so sd cap. , x - v / 2 ; y = 0;z = l maxP = 2<:; V i 15 do ta CO minP = 4 o x = y = z = l . Nhan xet: Ldi giai cua bai loan diJa vao linh binh d^ng gifl-a cac bien kel help :s. x = N/2;y-0;z-I. x^ + y^ =2 (4) iL ie uO nT hi Da iH oc 01 / O x=y=z=l I Dau bkng trong (2) xay ra o z - 1 (3) Dau bang trong (4) xay ra o dong thdi c6 dau bkng trong (2) (3) x=y (2) y-0 X + y + z + xyz = 4 o z ( l + xy) = 4 - (1) X -y oz=^-^-y 1 + xy (2) Ta CO 4 > X + y > 27xy , do d6 0 < xy < 4 V i X > y > 1 => X - 1 > 0; y - 1 > 0, nen theo ba't ding thffc Cosi, ta c6 x + y - 2 = ( x - l ) + ( y - 1) > 2 V ( x - l ) ( y - l ) => (x + y - 2)^ > 4(x - l)(y - 1) > xy(x - l)(y - 1) =>(x + y - x y ) ( x y + l ) > ( 4 - x - y ) ( x + y - 1 ) => X + y - xy Tff (2) (3) suy ra 4-x -y •(x + y - 1). xy + 1 X + y - xy > z(x + y - 1) =>x + y + z > x y + yz + zx =>P>0. (do xy < 4) ' • (3) Chuy6n 6i BDHSG Toan gia tr| Ifln nhS't 2. N c u X > x = y = z= 1 1 > y > z. K h i do ta c6 (x - l ) ( y - l ) ( z - 1) > 0 => xyz - (xy + yz + zx) + X Cty TNHH MTV DWH Khang Vigt gia tr| nh6 nha't - Phan Huy Khii X Tuf do ta C O maxP = 2 <=> + y+ z- 1> 0 mot so biing 1, mot so bang \f2 . Theo bat dang ihiJc Cosi, ta c6 4 = x + y + z + xyz > 4:^xyz(xyz) =>0 0. ( x - l ) ( y - l ) ( z - l ) = {) x = y = z = xyz T i m gia trj nho nhat ciia bieu thiJc (4) <=> X t,,,, j. P = iL ie uO nT hi Da iH oc 01 / X Da'u b^ng trong (4) xay ra <=> = y = z = 1. K e t hdp l a i ta c6 minP = 0 <=> x = y = z = 1. Dalx = HUdng ddii giai ' (D ' om /g "x = 0 ok 2 ^2 X , „2 + Z 2 y + + Z + < bo X 1 + Z 2\ w. , ..2 + Z 2 ww ,^2 T X X (4) x^+y^+z^ ZX XY Y^Z^ x^ + 2x + 4 4X'' + ylr^l 4X^ YZ , + 4 X"* + X ^ Y Z + Y ^ Z ^ Y4 X ' + X ' Y Z + y^Z^ l3 X % = 1 (5) x^+z^ O x = 0;y = l;z = (2) Z^X^) Y' + Z' + (X'Y' + Y'Z' + Z'x') + (X'Y' Z'x') + Y'Z' + => ( X ' + Y ' + Z ' ) ' + z'x') + X Y Z ( X + Y + Z) :;: (3) X = Y = Z. (4) (chu y P la long cua ba phan so dffdng) Dau bang trong (4) xay ra d6ng thcfi co dau bkng trong (3), (5) "x = y - z = l (1) Z ^ + Z ^ X Y + X^Y^ Tff (2) (3) suy ra P > 1 (6)' o V Y^ + Z^ + X Y Z ( X + Y + Z) + (X^Y^ + Y^Z^ + (X' + Y' + Z')' = X U 2 D a u bkng trong (6) xay ra ,.. .... Y^ + Y ^ Z X + Z - X ' Dau bang trong (3) xay ra o y' = f ^ I Tff do b^ng phep tinh Iffdng Iff, ta co > X ' + Y " + Z ' + ( X ' Y ' + Y'z' Dau b^ng trong (5) xay ra o X^ iTa co: (do x ' + y ' + z ' = 3) Tff (4) (5) suy ra P < YZ i D a u bang trong (2) xay ra o X = Y = Z fa Theo bat d^ng thufc Cosi, ta co 2Z' ( X ^ + Y ' + Z ^ ) ^ ce Tilf (2) (3), ta C O P < y ( x ' + z') = 2^ 2Y' | T f f (1) va theo bat dang thffc Svac-x(t, ta co: .c y= x y= z 2 ro (3) up (2) Tiir (1) va do X > 0 => x(y - x)(y - z) < 0. ^ s/ = xy^ + yz? + z x ' - xyz = ( y z ' + x ' y ) + ( x y ' + z x ' - x V - x y z ) = y ( x ' + z') + x(y - x)(y - z) ^ Ta V i e t lai P dffdi dang sau: 2X' V d i phcp d d i bien nay thi 4X^ nen khong mat tinh tong quat co the gia siJ y la so c( giffa x va z. Dau b^ng trong (3) xay ra <=> z^+2z + 4 Khi do la co X , Y , Z la ba so thiTc khac khong. Do vai tro cua cac so x, y, z trong bieu thufc P co tinh hoan v i vong quanh, P y^ + 2y + 4 Do xyz = 8, nen ta co the d d i bien nhu" sau: T i m gia trj Idn nhat cua bieu thufc P = xy^ + yz^ + zx^ - xyz. Nhirvay t a c 6 ( y - x ) ( y - z ) < 0 . x^+2x + 4 HU('fitg d&n gidi j B a i 32. Cho x, y, z la ba so thiTc khong a m va x^ + y^ + z^ = 3. < w . > * * . S - : S hoac la trong ba so x, y, z co mot so bang 0, + y + z - (xy + yz + zx) > 1 - xyz. Tir do suy ra P = . , r- V2. ; ^'^ VayminP= 1 o x o ddng thdi co dau bang trong (2) (3) o X = y = z= 2 = Y = : Z < = > x = y = z = 2. , ,. ^ , Cly TNHH MTV DWH Khang Vijt Chuyfin 06 BDHSfi Tpan gia tr| lOn nhiit va gii trj nh6 nha't - Phan Huy Khii Nhdn xet: Trong bai loan tren ta da ap dung phep doi bien day hieu qua sau day: pal 35. Cho x, y, z la ba so' thifc khac 1 va Ihoa m a n d i l u k i e n xyz = 1. \ V d i ba so' thiTc khac khong x, y , z thoa man dieu k i c n xyz = k"* thi c6 the ap Tim gia tri nho nha't ciia bieu thiJc P = dung mot trong ba phep d o i bien sau: ' kx ks/x CO Do xyz = 1, nen thiTc hi?n ph6p d o i bien sau ( x e m phep d o i bien thur trong kx I = X. nhan xet cua b a i 33) X' Vay tiTcfng tiT c6 y = kY^ kZ^ ,z = Z X ' " XY kX^ kY^ kZ^ -;y = -;z = YZ ' ZX XY 2. Dal X = 1 „ 1 . kYZ kZX kXY (ban doc -,z - • tif nghiem lai). tV) .-.Ml r 2 r x^ ^ YZ VYZ + 2 ZX {X'-YZf XY + Izx ; 7} Y ^ ^ ) I X Y (Y^-ZXf ) {Z^-XYf TO (1) va Iheo ba't dc^ng thiJc Svac-xd, ta c6: ^ ^ ^ ^ ^ kY kZ kX 3. D a t X = -n= '- Y = T7= ; Z = -7= hay x = — ; y = — ; z = — 3/7 3/7 3/r ^ 7. Y Y up s/ Ta ^ Tim gia trj Idn nha't cua bicu ihiJc P = z , jj. om HUdngddngidi X ok Ap dung phep d o i bien thi? ba trong nhan x e t ciia b ^ i 33 ( v d i k = 1), cu the X Y Z Y Z X (X^ + Y ^ +Z^)^ P> (X^ -YZf +{Y^ -ZXf i P a u bkng trong (2) xay ra « X = Y = Z. ' ^ (2) " ! fa CO (X^ + Y ^ + Zy = X " + Y ^ + Z^ + 2(X^Y^ + Y^Z^ + Z^X^) J . , , |.:.:(X^- Yz:>^ + ( Y ^ - z x ) ^ + ( z ' - X Y ) ' ( X ' + Y ' + Zy- |(X- - ce fa w. ww HUdng ddn giai - •< YZ Dau bkng trong (3) xay ra <=> X = Y = Z o X = y = z = 1. '* JJ^u • B&i 36. Cho X, y, z la ba so thiTc difdng va thoa man dieu kien xyz = 1. 1 1 1 Tim gia tri nho nha't cua bieu thiJc P = 7l + 8x Vl + 8y Vl + 8z • XYZ > (X + Y - Z)(Y + X - Z)(Z + Y - X). " >: '/ YZ)' + ( Y ' - ZX)' + (Z' - XY)'] Do X > 0, Y > 0, Z > 0, nen ta CO ba't dang thtfc quen bie't sau day '^'^'^ q ', ^ Vay minP = l<::>x = y = z = I . XYZ Dau bhng trong (2) xay ra o X = Y = Z. .:,,„,„1, I Dau bang trong (3) xay ra o X = Y = Z = 1 o x = y = z = 1. fY X , Luc d6 bieu thtfc P c6 dang P = —+ 1 X X U Y Z z (X + Z - Y ) ( Y + X - Z ) ( Z + Y - X ) Vay ta c6 max? = I o x = y = z = 1. ' ca mau va tuT cua phan so vc' phai dcu diTdng, nen tir (2) suy ra P > 1. X = — ; y = — ; z = — k h i do ta c6 X, Y, Z la ba so thiTc diTdng. T i r ( l ) ( 2 ) s u y r a P < 1. +(Z^-XYf | r (XY + YZ + Z X ) ' > 0. bo dat I = X " + Y ^ + Z % X ' Y ' + Y ^ Z ' + Z ' X - - 2 X Y Z ( X + Y + Z) .c y /g ro Bai 34. Cho x, y, z la ba so' Ihi/c di/cfng thoa man dieu kicn xyz = 1. • Z I Luc nay bieu thufc P c6 dang P = (trong ba: tap tren da dung phep thay bien nay vdi k = 2) , Y^ x = TYZ 7;r;y = — ZX; z = - XY Nhifthc'co the vie't lai phep thay bien thanh x 1 ' iL ie uO nT hi Da iH oc 01 / • J ' . Khi do ta z-1 HU&ng ddn giai 1. D a t X = ^ ; Y = ^ ; Z = ^ T^i + y-1 ^''^ 1 ZX Do xyz = 1, nen thi/c hien phep doi bien sau x = —5-; y = — r2 -' \ = X^ XY d day X, Y, Z la cac so diTt^ng (xem phep doi bien thur hai trong phan nhan xet cua bai 33 vdi k = 1) Khi do bieu thtfc P trd thanh -vo ^-^^^^ ^ • 181 Cty TNHH IVITV DVVH Khang Vi^t Chuy6n dg BDHSG Toan g\A trj Wn nha't va gia tri nh6 nhat - Phan Huy Kh^i 1. Cho x, y, z la ba so du'dng vii xyz = 1. , . , . , . a. T i m gia tri nho nhat cua bieu thUc P = x+3 ir (x + X Y Z VX^+8YZ VY^+8ZX VZ^+8XY y+3 +— + {y + \f z+ 3 (z + 1) 2 • D a p so': minP = ^ . , ^^^^^ ^^^^ b. T i m gia trj k'ln nhii't cua bicu thu'c: +8YZ Y V Y ^ +8ZX z7z^+8XY Tit (1) va Iheo ba't dang thuTc Svac-xd, ta c6 / ' , p. , Q= (2) XVX^+8YZ + Y>yY^+8ZX+ZVZ^+8XY +21y + 9 Ta up T i m ilia t r i Idn nhat cua bieu thu'c: 1 1 - xy om ok bo + Z(X - Vf > O. (6) dan 1 - zx giai Neu 1 + 9xyz - X - y - z < 0 . * Neu 1 + 9xyz - x - y - z > 0 . Theo bat dang thtfc Cosi, ta c6: x + y + z = (x + y + z). 1 = (x + y + z)(x^ + y^ + z^) > 37xyz-3>/x^y^z^ ^ ^, = > X + y + z > 9xyz = > l + x + y + z > l + 9xyz (7) Da'u bang trong (7) xay ra <=> X = Y = Z <=> x = y = z = 1. =>0< 1 + 9 x y z - X - y - z < 1 => (1 + 9xyz - X - y - z) ^ xet: 1. Qua cac bai 33 - 36, cac ban da tha'y ro h i c u qua to Idn cua cac phep doi bic'n (da trinh bay trong nhan xet cua bai 33) de giai nhieu bai loan t i m gia . 1 1 1 - xy V a y minP = l o x = y = z = l . Nhan 1 - yz =>P<0. (5) Do X , Y, Z > 0, nen (6) dung va dau bang trong (6) xay ra c > X = Y,= Z T i r ( 4 ) v a ( 5 ) c 6 P > 1. 1 +• Do x^ + y^ + z^ = 1 => |x| < 1, |y| < 1, |z| < 1 => 1 - xy > 0, 1 - yz > 0, 1 - zx > 0 fa Xr w. + Y(Z - * ww Zr ce Cothe thayrang(X + Y + Z ) ' > X ' + Y ' + Z ' + 24XYZ 1 •+ /g ro P = (1 + 9xyz - x - y - z) c~> ddng ihcti c6 da'u bang Irong (2) (3) < ~ > X - Y = Z o x = y = z = 1. <• « 1% HUdng .c VX^ + Y ^ + Z- + 2 4 X Y Z Vz^+21z + 9''''-*y Bai 37. Cho cac so thifc diTdng x, y, z thoa man dieu k i e n x^ + y^ + z^ = 1. s/ (3) / X + Y + Zt^ Tir (2) (3) suy ra P > ' 7(X + Y + Z)(X-^ + Y-^ + Z-^ + 2 4 X Y Z ) 1 Dap so': minP = ^ . ...rfn.'/U < V(X + Y + ZHX"* + Y-^ + Z-^ + 2 4 X Y Z . (5) o X ( Y - f ''"X'-it'Oilt,.-:': ^'^y^ + Z + 4 1 Vx^+21x + 9 That vay bang cac phcp b i c n d d i sd cap, la c6 1 + - \l4y^~+y^^ 1 p = + 8 X Y Z + V Y VY-^ + 8 X Y Z + N/ZVz'^ + 8 X Y Z Dau bang trong (4) x i i y ra 1 T i m gia t r i nho nhat cua bieu thu'c: + 8 Y Z + Y ^ y ^ + 8ZX + zVz^ + 8 X Y Vx-^ V4x^ + X + 4 + 2. Cho X , y, z la cac so' difdng va thoa man dieu k i e n xyz = 27. Thco bat dang thu'c Bunhiacopski, ta c6: = VX . D a p so: maxQ = 1. Dau bang trong (2) xay ra o X = Y = Z xVx^ 1 iL ie uO nT hi Da iH oc 01 / xVx^ d: Da'u bang trong (1) xay ra 1-yz 1 + ' l-zxy 1 < 1-xy <=> 1 + 9xyz - x - y - z = 1 o X + y + z = 9xyz <:ox = y = z = — 1 + 1 + 1-yz (1) 1-zx '' ' • " ' (dox^ + y^ + z^= 1). trj idn nhat va nho nhil't bang phu'ring phap bat dang thu'c. 2. Cac ban hay ap dung cac phcp doi bic'n ay de giai cac bai toan sau: 183 Cty TNHH MTV D W H Khang Vigt ChuySn (SJ BDHSG Toan g\i tr| Idn nhgt va gia tr| nh6 nhgt - Phan Huy KhSi ' ^' Neu m > 0 , n > 0 t a c 6 — + — > va da'u dang thu'c xay ra m n m+n a b ta c6 <=>—==-; m n x2 y2 (x + y)^ (x^+z^) + (y^+z^) \ ^ + z ^ ' y^+z^ ' yL.' Tilf (3) (4) suy ra ^ ^ < i 1-xy 2 Dau bang trong (3) xay ra o (4) om .c ok bo < -+ 1-yz 2 y^ + x^ z^ + x^ zx 1 z^ •+ • 1-zx <-2 Tir(2)(5)(6)(7)suyra Q<3+| = |. >, ce ,2 ww Baygidtir(l)(8)c6Q< - . 2 Dau b^ng trong (9) xay ra de tha'y khi x = y = z = 9 <=>x = y = z = 73 Ket hdp ca hai triTcfng hdp suy ra maxQ = — —. ••• ,r? Bai 38. Cho ba so thifc x, y, z e [ 1; 2] 1. Tim gia tri Idn nha't ciia bieu thu'c P = (x + y + z) (5) (6) (7) (8) w. TiTdng tif ta c6 y2+z2; fa ^x^+z2 iL ie uO nT hi Da iH oc 01 / • l-iii rf. 1 - P —+ —+ — ,x y zj Ta ^ y HUdng dan gidi 1 VietlaiPdi/didangP=- + ^ + - + ^ + - + - + 3. y X y z z X „ _do ta , CO x -y < z1; - y< 1.x Viz vay suy ra ' 1 - Tiir >0 yj X y y X y z z X i >. . •>••• z Do vai tro binh dang giiJa x, y, z ncn c6 the gia suf x > y > z. X y X ; •:>. m:. s/ 2 (2) up 2 U ro 2 1 1^ 2 Tim gia tri Idn nhaft cua bieu thu'c Q = (3x + 2y + z) - + - + f1 /g D a t Q = —i—+ —1—+ — k h i d o : 1-xy 1-yz 1-zx Q - J~xy + xy ^ l - y z + yz ^ 1-zx + zx 1-xy 1-yz 1-zx ^ xy ^ yz ^ zx =3+ 1-xy 1-yz 1-zx^ Tilf gia thiet x^ + y^ + z^ = 1, ta c6 (do x^ + y^ > 2xy) 1 xy < 2xy (x + yr 1-xy 2 - ( x ^ + y ^ ) . < 2 (x^ +z^) + (y^ +z^) (do x^ + y^ + z^ = 1 va (x + y)^ > 4xy). Ap dung ket qua sau day: (9) 1 =1 z , x-y V =z (1) (2) (3) ^ .y Lai CO - > 1 , ^ > 1 => 1 - - 1-y > 0 X y y z , X y 1 • (4) =:>- + - < l + - . Da'u b^ng trong (4) xay ra o x=y o y = z. y =l z C6ngt£rngve(3)(4) vkco - + ^ + - + ^ < 2 + -X + -z . *i v : • y (5) y z z x Dafu bang trong (5) xay ra o dong thcJi c6 da'u b^ng trong (3) (4) 'x^y _y = z. Mat khac tCf gia thiet c6 1 < x < 2 ; l < z < 2 = > x - 2 z < 0 ; z - 2 x s 0 => (X - 2z)(z - 2x) > 0 , . (6) X z 5 oDau2x^bhng + 2z^ ^ 5xy . trong (6)oxay- ra+ - o< -x=:2z z=:2x z X 2 185 X Chuyen dg BDHSG To^n gia trj I6n nhS't va glA tr| nh6 nha't - Phan Huy Kh^i Cty TNHH MTV DVVH Khang Vi^t B a y g i d tit (5) c6: Ttf(l)(2)suyraP>2. R = - + - + - + - + - + -<2 y X y z z X +2 <2 + 2 . - = 7. 2 (7) (4) D2'u bang trong (4) xay ra <=> x + y + z = 0 <=> x + y = - z ^ x + y = x +y D a u bang trong (7) xay ra o dong thcfi c6 dau bang trong (5) (6) o x =y y =z <=> y, z G trong ba so x, y, z c6 2 so bang 2,1 so bang o T i m gia t r i Idn nhat cua bieu thifc: trong ba so' x , , z co 2 so bang 1,1 so b^ng 2 P = z <—(x + y + z) 4 1 1 1 —+ —+ - K e t h d p p h a n 1, 2 s u y r a Q < - . 1 0 = — . /g ^ ; D a u bang trong (8) xay r a o y + 5z = 6 = 3 x c : > x = 2 ; y = z = l . : ^- . I 2 om S (9) .c 4 Ta s/ - up 1 Da'u bang trong (9) xay ra o dong ihdi c6 dau bang trong (7) (8) - i ± ^ x +y xy + yz + zx = - 1 . K h i do P = x ' + y^ + 1 + xy x+y ISA 1-X Tir (1) suy ra X {;•:: (x + l ) ( y + l)(z + l ) ••'0 ••^ ' 1+z 1-Y ,I . Y "l*-* ;< + | 1-Z ;y= = ^ 1 y. (1) ^ ;z = l +Y +X 1 +Z R6 rang tir xyz = 1, suy ra (1 - X)(l - Y)(l - Z) = (1 + X ) ( l + Y)(l + Z) ^ x + Y + z + xYz = o. _ i L _ = i _ x ^ - ^ = l + X, y+i • " /v,v. , . = l - Z ^ - ^ = l + Z. (z +1)^2 '• if&il ' , z +. 11 V i vay P CO dang P = - ! - [ 3 - ( X 2 + Y ^ + Z ^ ) - 2 ( l + X)(l + Y)(l + Z)] 4 (1) (2) = 1J3 _ [(x2 + Y^ + Z^) + 2(XY + YZ + ZX) + 2(X + Y + Z + XYZ) + 2 ] (3) i r .1 - ( X + Y + Z) ^,2 4L J 4 Tiir (2) (3) suy ra P = - + 2(xy + yz + zx) > 0 => x^ + y^ + z^ > - 2 ( x y + yz + zx). ;Z= (y + 1) - R6 rang ta c6 (x + y + z)^ > 0 =:> x^ + y^ + 4 (z + 1)^ 1+y >'•' ^ ^ ' ( X + l)2 X + 1 ^ = i - Y ^ - i - = i + Y . Hiidng ddn gidi D3tZ = ^s...^ ce ww B a i 39. Cho x, y la cac so thifc va x + y ;t 0. T i m gia t r i nho nha't cua bicu thiJc P = x^ + y^ + (y + 1)^ • Do X, y, z deu la cac so diTcfng, nen de thay X, Y, Z e ( - 1 ; 1) fa <=> x = 2; y = z = 1. w. VaymaxQ= — ' """^'h z 1+x Taco ok = z= l;x = 2 (x + 1)^ bo ' o y 45 y DatX= 1::^; Y = ro 1 x " Hiidngddngiai 9 9 => - ( x + y + z) - (3x + 2y + z) > 0 => 3x + 2y + z < - (x + y + z) 4 4 1 ,. Bai 40. Cho x, y, z la ba so thiTc du'dng va thoa man dieu k i e n xyz = 1 . 11; 2] =:> y + 5z > 6 > 3x. —+ —+ X y (5) (1)), va diTa vao m o t ba't dang thiJc cho triTctc dc giai b a i toan dat ra. maxR = 7 hay maxP = 10. • (3x + 2y + z) : Vay minP = 2 <=> x, y thoa man (5) 9 1 2. T a c o - ( x + y + z ) - ( 3 x + 2 y + z) = - ( y + 5 z - 3 x ) . 4 4 X, ^ pinh ludn: 6 bai trcn ta da silr dung k l ihuat: Diing mot dang thiJc (dang ihi^c ' ' Gia t r i Idn nha't dat diTdc Do • That vay chang han x = 1; y = 0 thoa man (5). (dox>y>z) ^ y = z = l;x = 2 X = 2z hoac z = 2x T o m l a i ta c6 -''i^ Chii y rang tap cac so thiTc x, y thoa man (5) la khac rong. = y = 2; z = 1 X iL ie uO nT hi Da iH oc 01 / X = 2z hoSc z = 2x (x + y)- = 1 + xy. (3) Dau bhng trong (4) xay ra o X +Y +Z =0 10*7 Chuy6n (SJ BDHSG Join g\A trj lan nha't va g\i tr| nh6 nha't - Phan Huy KhSi Cty TNHH MTV DWH Khang Vi?t 1-x 1-y 1-z ^ + — - + =0 1+x 1+y 1+z Ttf (1) ta CO (X + y)(y + z)(z + x ) > 3(x + y + z) ^ ^ y2.,2^2 ^ z ^ - xyz , (X + y)(y + z)(z + X) > ^ / x V z ^ ( 3 ( x + y + z ) - ^ / ^ ) . xyz = l R6 rang tap cdc so thuTc difdng x, y, z thoa man (5) la khac (thi du c6 the lay x = y = z = l ) . V a y minP = i o x, y, z thoa man (5) ' Ttf (2) (3) suy ra 1. Trong b a i nay ta cung suT dung mot dang thiJc va mot bat dang thtfc de g i j j bai toan. Cho X, y, z la cac so di/dng va thoa man dieu k i ^ n xyz = 1. T i m gia t r i nho nhat cua bieu thiJc: 1 + zr ( l + x ) ( l + y ) ( l + z) ' ,^ . ' ! A 1-X 1-Y 1-Z y = -——; z = 1+X 1+Y 1+Z 1+X i 1)5 1+Y , y . . : ' up 2 2 ; 1+ z = 1+ Z + (1 + X ) ( l + Y ) ( l + Z)" (6) .c N h i / v a y P = - [ ( 1 + X ) ^ + (1 + Y ) ^ + (1 + Zf 4L (4) ce , , w. Vay ta CO P > 1 => minP = 1. ' ' Difa vao dong nha't thiJc: 2. Ta c6 bai toan tiTdng tiT sau: Cho X, y, z la cac so thifc diTdng. T i m gia t r i Idn nha't cua b i e u thtfc P = 1+ ^ l+y Tijf do theo bat d i n g thiJc Cosi, ta c6 x y + yz + zx > xyz * i < •• % i ' (x + y)(y + z)(z + x ) 2(x + y + z) ^/xyz (1) '^ 3 ^/xyz (x + y ) ( y + z)(z + x ) xyz (1) „ -w* 3^/?77. ' (x + y ) ( y + z)(z + x ) ^ 8 x + y + z ^ ((1) chi^ng m i n h de dang v^ x i n danh cho ban doc) Da'u bang trong (2) xay ra<=> x = y = z = l . 2(x + y + z) 1+^ 8 \ Theo b ^ i tren ta c6 (x + y ) ( y + z)(z + x ) > - ^ x ^ y ^ z ^ (x + y + z) t (x + y ) ( y + z)(z + x) = (x + y + z)(xy + yz + zx) - xyz ,, : ^ t xyz i ' = y = z = 1. f T i m gia t r i nho nha't cua b i ^ u thiJc P = (x + y ) ( y + z)(z + x ) x +y+z HUdngddngidi dong thcJi c6 da'u bang trong (2) (3) 1. M a u chot la diing dong nha't thuTc (1) ww Bai 41. Cho x, y, z la ba so' thifc diTdng va thoa man xyz = 1 . • '' V i e t l a i P diTdi dang P = u.,./, fa ^ ( HUdngddngidi bo Giong n h i r b a i tren ta CO X + Y + Z + X Y Z = 0 1 Binh luqn: T i r d o de dang suy ra (1 + X)^ + (1 + Y)^ + (1 + Z)^ + (1 + X ) ( l + Y ) ( l + Z ) > 4 (ban doc tif chtfng minh l a y ) . o M Vay minP = - <=> x = y = z = 1. . ( ro ; 1+ y = x +y+z (5) (do x y z = l ) ~ 3' o x - — — ; 2 3(x + y + z ) - (x + y)(y + z)(z + x ) ^ 8 x +y+z /g 1 + X= > Da'u bang trong (5) xay ra X = ^ - ^ - Y ^ ' - ^ ; Z = ' - ^ , 1+x l+y 1+z \ I X = x) > ^ / x V ? • Ta (1 1 _^p^ •); .» s/ ( l + y) N om => + xr r +- + y ) ( y + z)(z + ok Dat (1 1 (x . ' o i • (X + y)(y + z)(z + X) > - ^ x ^ y ^ z ^ (x + y + z) 2. X e t bai toan tifcfng tif sau: ^ + x = y = z = 1. iL ie uO nT hi Da iH oc 01 / A^/ia/i xet: 1 (3) i Oa'u bang trong (3) xay ra o „ P= x +y+z L a i theo ba't d i n g thuTc Cosi, ta c6 ^/xyz < 1 2x +y+z^^x +y +z 3 (x + y)(y + z)(z + x ) — ________ xyz t xyz xyz 2(x + y + z) ^ 2 x + y + z ^ _ 3 (2) L a i CO ^ t J ^ ^ > 3 (theo bat d i n g thtfc Cosi) ^xyz ^^'xyz (2) Chuy6n dg BDHSG Toan g\i trj Idn nha't va gia trj nh6 nhat - Phan Huy KhSi P>2 ^ .• ,., PHUaN6PHliPllf9N6GttCHdA 0,^^ Vay minP = 2 <=> X = y = z > 0. TiMGlATRIltfNNHKtlNi NHiNHlttCdAHAMStf B a i 42. Cho x, y, /. > 0 va ihoa man xy/ - 1 1. T\m gia trj nho nhat cua P = (x + y)(y + /.)(/. + x) - 2(x + y + /). fx + y ly + z Iz + x 2. Tim giii tri nho nhat ciia hicii thiJc Q = . + + , V x + l \ y + l V z + l ffUihtgddn . ^Kt j :^>f? gidi iL ie uO nT hi Da iH oc 01 / „ /V . . . y LiTdng gi^c h6a I I m o t trong nhi^ng phrfdng phdp hay suT dung de t i m gia t r i '' Idn nhaft, b6 nhaft cua h a m so'. 1. A p dung dong nhat thufc: B^ng phifdng phap d d i bien lifdng giac (thi du x = sint, x = cost h o l e x = + y)(y + z)(z + x) = (x + y + z)(xy + yz + zx) - xyz. (*) Ta c6: P = (x + y+ z)(xy+ yz + z x ) - x y z - 2 ( x + y + z ) . tKt" (X T h c o ba't dang thiirc Cosi, ta c6 x + y + z > 3 ^/xyz L a i CO xy + yz + zx > 3 ^ / x V z ^ = 3 (do x^yV^ TCr ( I ) (2) (3) suy ra P > d) ,.>,Vr* - + y + z) - 1 > 3 - 1 = 2. Ta (4) ,;. : up y)(y + z)(z + x) > (X + l)(y + l)(z + 1) ro (3) /g om .c bo ok , , , x y + yz + zx (7) + (x + y + z)(xy + yz + Ho$c la dieu k i e n trong bai toan ban dau c6 dang: x^ + y^ = a^ a > 0,... - HoSc la cdc bieu thtfc da cho ban dau g^n lien v d i m o t h? thtfc liTdng gidc quen biet nao do. T i m gia t r i Idn nha't cua bieu thiJc P = ^ x y z + J ( l - x ) ( l - y ) ( l - z ) . Htidng ddn giai Do x, y , z G [ 0 ; 1 ] , nen dat x = sin^A, y = sin^B, z = sin^C, zx) r—^ ' («) Tilf (7) (8) suy ra (x + y + z)(xy + yz + zx) > x y + yz + zx + x + y + z + 3. K h i d 6 0 < s i n A < l ; 0 < s i n B < l , 0 < s i n C < l ; 0 < c o s A < 1; 0 < c o s B < l , 0 < c o s C < 1. ' ' r'I iv Ta c6: cosAcosBcosC 3. Vay minQ = 3 o x = y = z = l . ! Ldc nay ta c6: P = Vsin^ A s i n ^ Bsin^ C + Vcos^ A c o s ^ B c o s ^ C V a y (6) dung, tuTc (5) dung. Theo bat dSng thtfc Cosi, ta c6 Q > 33 - vdi A, B, C e ww ( x y + yz + zx) + (x + y + z ) - ^ - 4 w. x +y+z, ^ fa Ta CO (x + y + z)(\ + yz + zx) chlfa cdc d a i liTdng dang x^ + y^; 1 + x^;... (6) ce D o xyz = 1 = > X + y + z > 3 va x y + yz + zx > 3. ^ z + 2 (do xyz = 1) + y + z)(xy + yz + z x ) > x y + yz + zx + x + y + z + 3. HoSc la trong bieu thtfc cua d a i liTdng can t i m gid t r i Idn nhat, nh6 nha't c6 B j k i l : C h o x , y , z G [ 0 ; 1]. (5) <=> (x + y + z)(xy + yz + zx) - xyz > xy + yz + zx + x + y + z + 1 \ " o (x + y + z)(xy + yz + zx) - 2 > xy + yz + zx + x + y + bai toan t i m gia t r i Idn nha't, nho nha't c6 the suT dung phtfdng phdp liTdng gidc hoa thi/dng c6 cac dau hieu de nhan bie't sau day: 3(x + y + z) - 1 - 2(x + y + z) That vay dufa vao (*) suy ra = Cic (3) 1) 2. TrU'dc he't ta chii'ng minh rhng o(x tim gia t r i Idn nhat, nho nhat da cho ban dau. (2) = 3 (do xyz = 1 ) De thay dau bang trong (4) xay ra <=> x = y = z = 1. (x + difa vao phep tinh Itfctng giac ta se de dang hdn trong trong viec g i a i b a i toan s/ => P > (X tant,...) ta diTa bieu thiJc va dieu k i e n cua bai todn ve dang luTcJng gidc. Tir d6 o rx = l (*) LLy = l O) Chuy6n gj BDHSG Toan glA trj Idn nhat va g i i tri nh6 nhS't - Phan Huy Khii sinC = l Dau b^ng trong (3) xay ra <=> sinAsinB = 0 Cty TNHH MTV DWH Khang Vi$t => tan^a tan^p tan'y tan^ 8 > 81 => x''y''zV > 81 => P = xyzt > 3. 'z = l <=> x=0 Dau bSng trong (7) xay ra <=> dong thcJi c6 dau b^ng trong (4), (5), (6) (**) <=>a = P = Y = 8 y =o Tir (1), (2), (3) c6: P < cosAcosB + sinAsinB =>P< cos(A - B). ^ . V i cos(A - B) < 1 va dau b^ng xay ra khi va chi khi A = B, nen ta c6: ^hdn xet: Hoan toan tiTcfng tif, ta co ke't quS sau: C h o x > 0 , y > 0 , z>Ovlk —^—- + —J—+ — + 1 + x^ 1 + y^ l + z" dong thfJi thoa man (*) va (**) dong thdi thoa man (*) (**) (6) ^ Ta s/ up ro om ok bo fa Tim gid tri idn nhat va nho nhat cua ham so: f(x) = ^ + 4x + 3x (i+x^r (1) HUdng ddn giai (2) (3) sin^p > 3^cos^acos^8cos^^ y , (4) sin^y > 3^os^ a cos^ Pcos^ 8 , (5) sin^ 8 > 3^/cos^ a cos^ pcos^ y . (6) sin^asin^PsinVin^ 5 > 8 Icos^a cos^p cos^os^ 8 - B^i 5: Cho x la so thifc tily y (x e R). sin^a >3^/cos^8cos^cos^y . Nhan tiTng vd' (3), (4), (5), (6) va c6: Bai 4: (De thi tuyen sinh Dai hoc, Cao ddn^ khoi B) . • o Tim gia trj Idn nhat \h nho nha't cua ham so: f(x) = x + \/4-x^ tren mien Xem Idi giai each 3 trong bai toan 1, bai 1, chiTcfng 1 cuon sach n^y. w. ww Tir (1) suy ra: sin^a = cos^p + cos^y + cos^6 Xem IcJi giai trong bai toan 3, § 1, chiTdng 1 cuo'n sach nay. Ddp so: max f(x) = 2>j2 ; min f(x) = - 2 ce cos^ 8 (; — ^ = 1. 1 + t^ Hudngddngidi VI vay dieu kien: + -r + j + j =1 • 1 + x^ 1 + y^ l + z^ 1 + Lap luan tiWng tif, ta c6: , xdc dinh cua no. .c I + y" = — 1 - • cos'p 1 /g Dat x^ = tana; y^ = tanP; z^ = tan y; t' = tan 8 vdi a, P, y, 8 e ' Ap dung bat dang thiircCosi, ta c6: , Hitdng din giai Dap so: max P = 3; min P = - 6 Hudngddngidi o cos^a + cos^p + c o s \ cos^ 8 = I , Chox^ + y^ = i . • '(pp!< Tim gia trj nho nha't cua bieu IhiJc P = xyzt. cos^ y . Tim gia tri Idn nha't va nho nha't cua bi^u thtfc: P = ^ ( ^ + 6 x y ) 1 + 2xy + y^ Bai 2: Cho x, y, z, t > 0 va thoa man dieu kien: 1 1 1 1 T + T + 7 + 7 =1 • CQS Qt ; Bai 3: (De thi tuyen sinh Dai hoc. Cao dann khdi B) Tir do ta CO max P = 1 o x, y thoa man (6). Tilfd6: 1 + x" = 1 + tan^a = - 4 — ; - .; Khi do neu P = xyz, thi min P = 2 72 . x=y=z=l -'^ (8) < ' . , iL ie uO nT hi Da iH oc 01 / Dau b^ng trong (4) xay ra o dong th5i c6 dau b^ng trong (2), (3). z - 0 ; x = 0;y = 0 ' « x = y = z = t= (4) Vay min P = 3 o x, y, z, t thoa man (8). P<1 (5) Dau bkng trong (5) xay ra o A = B <=> X = y (7) E>^tx= tancp vdi x e "2' 2 Khi 66 ta c6: 3 + 4x^+3x^ 3 + 4tan^(p + 3tan'*(p , , ^ 4 \ — "2— = 5—21=:(3 + 4tan''(p + 3tan^(pjcos 9 ( l + x^) ( l + tan2(p) i - - s i n ^ 2 ( p + sin^2(p ' • = 3cos'*(p + 4sin^(pcos^(p + 3sin'*(p = 3 \ ' 2 = 3-—sin^2cp. (1) CtyT.lli M i V DVVH Khang Vi$t ChuySn BDHSG Toan gii trj Idn nha't g\& tr| nh6 nha't - Phan Huy Khii X6t ham so F((p) = 3 - ^sin^ 2(p, vdi cp e Tir n 7C do: f'(x) = (8x + 12x'')(l + x ^ ) ^ - 4 x ( l + x^)(3 + 4x2+3x'*) (l + x '2''2) ^(8x + 12x-'')(l + x ^ ) - 4 x ( 3 + 4x^+3x^) Ta tha'y ngay: min F(cp) = 3 - ^ = | « si 2(p = 1; (l + x^)' Vay C O bang bien thien sau: maxF((p) = 3 - 0 = 0 o s i n ^ 2 ( p - 0 . . min De thay: 2'2, * ' nen tijf bang bien thien tren suy ra: 2x^ Ta 3x'* +4x^ +3 Goi m la gia tri tuy y cua f(x). Khi do c6 phufdng trinh sau (an x) 3 + 4x^+3x^ 1 + 2x2+x'* /g Vay maxf(x) = 3<:>x = 0. om xeR bo (4) ww w. • Tir do va theo (5) suy ra: min f(x) = | <=> x = ±1. 2 Ta thu lai ket qua tren. 2. Ta lai c6 c^ch giai khac nffa (bkng phiTdng phap chicu bien thien h i m so) -^^ '^^ f ( l + x^) . X€M. ' Vay m = 3 la mot gia tri ciia f(x). Dau bang trong (4) xay ra o x^ = 1 o x = ± 1. Taco: f(x) = '"'^ Neu m = 3, khi do (6) co dang: 2x^ = 0 o x = 0 (3) B a y g i 5 t i r ( l ) , ( 3 ) l a c 6 : f(x) > | V x e R . ^^i-^'' '> <=> (m - 3)x'' + 2(m - 2)x^ + m - 3 = 0 (6) VxeR. fa J Tir do theo (2) suy ra: — <-. x"+2x^+1 2 (5) (5) o 3 + 4x^ + 3x'* = m + 2mx' + mx"* (2) ce =>(X^+1)^ >4X^ .(Ivj = m nghiem. Do x ' + 2x^ + 1 = (x^ + l)^ >OVx,nen ok .c , ±1 Ta con c6 each giai khac niJa bang phiTdng phap mien gia tri ham so nhifsau: ro up s/ 1. Viet lai f(x) dirdi dang: f ( x ) - - 7 5 =3 — r — • x^+2x^ + 1 X +2x''+l Do >0 ly} „ Vx, nen iCf (1) suy ra: f f ( x ) < 3 V x e M lf(0) = 3 x''+2x^+1 xeR + ' 5 maxf(x) = 3 <=> X = 0; minl"(x) = - <=> x xeR xeR 2 thtfc) VXGR 0 ( l + x^) 2 X€R A^/ian jce^' X6t cdch giai khdc sau day: (bkng phUdng ph^p suT dung bat ding Dox^+l>2x +00 Chii y r^ng do lim '^^^^^ ^'^f ^ 3 maxf(x) = 3 o x = 0 ; minf(x) = ^ o x = ±1. xeR M a t khac: 1 f(x) XER (i + x ^ r •' f'(x) 2'2, . '^'^^ 0 iL ie uO nT hi Da iH oc 01 / 0 2(m-2) m-3 > ; ;l 2m-5>0 >0 » i m-2 m-3 <0 o — - < m < 3 . max P = 2 V 2 - 2 , min Tir do suy ra: max f(x) = 3; min f(x) = ^ . X£R xeR , P= -2V2-2. 2 Ttf (3), (6), (7) suy ra: , toan gia tri Wn nhat, nho nhat cua ham so. max P = max [ijl j min iL ie uO nT hi Da iH oc 01 / Sh$nxet: j DT nhien ta c6 cac ciich giai khac nhau nhifsau: Ta c6: Ta (2) max P; max P (x;y)eD| min P = min 1(1). (x;y)f.D2 ItK +2t + l va CO bang bien thien sau: (x;y)eD2 f(t) .2^[2-2. '0 om Khi (x; y) e D,, do y = 0 ncn luc do x i 0 va P = - — = 0 V(x; y) e D, => min P - max P = (). (3) (x;y)eD| ok {x;y)eDi .c X" X ce bo Khi (x; y) e D2, do y 9^ 0 nen luc do c6 the viel lai P diTdi dang: fa A _ 2 .2y. 4.1 *r 1 iTiirdosuyra: min P = - 2 > ^ - 2 v a (x;y)eD2 max P = 2 V 2 - 2 . (x, y)6D2 ,^ , -» .ll Pa thu lai kc't qua tren. , 2. Lai CO each giai bing phiTdng phap mien gia tri ham so: . 4t - 4 Goi m la gia tri tijy y cua ham so — , khi do phrfclng trinh (an t): (4) 4 t - 4 = (t^+ Dm (*) CO nghiem. Do (*) o mt^ - 4t + m - 4 = 0, nen de thay (*) c6 nghipm ; ww X w. P= +00 0 /g ro dday taco: D = D, UD2. 1 + N/2 1 - V2 s/ (x;y)6D (x:y)eD2 - r = 4 t ^' + ~ 1* (t^ + l ) ' (1) up max P = max leR De thay f'{t) = 4 Khi do Ihco nguycn l i phan ra, ta c6: (x;y)eD| max P = max f ( t ) ; (x;y)eD2 G o i D , = { ( x ; y ) : y = 0} vaD2= { ( x ; y ) : y ^ O } . (x;y)6D ' Dat — 2y = t, t e R. luc do xet ham so Rt) = ' l + t^ <- r . min P P=:min|-2V2-2;0|=-2N/2-2. (x;y)£D x^ — (x — 4y)^ Tim gid tri Idn nha't va nho nha't cua bieu thiJc: P = : 5—. x^ +4y'^ .: N HUdng ddn giai \ min P; - 2; o | = 2^2 - 2 , (x:y)eD Bai 6: Cho x va y la hai so thifc khong dong nhat bKng 0. min P = min (7) (x;y)eD2 ,, Mot Ian niJa cac ban thay diTOc tinh da dang cua cdc phUWng phap giai bai Dat D = { ( x ; y ) : x ^ + y ^ >0J (6) (x;y)eD2 7t_ 71 , ta c6: Dat — = tan a , vdi a 6 I '2'2, • 2y 2 p. ^ tan g - ( t a n g - 2 ) 1 + lan^ g Tir do thu lai ket qua tren!. Sai 7: (De thi tuyen sink Dai hoc, Cao ddnf; khfi D) 2 ^ ^^^^2 a-A) = 4sinacosa - 2( 1 + cos2a) -2. 4/ (5) , ; Cho X > 0, y > 0. Tim gia tri Idn nha't va nho nha't ciia bieu thuTc: p _ (x-y)(l-xy) = 2sin2g - 2cos2g - 2 = 2N/2 sin 196 <;#< : (l + x ) ' ( l + y 2 ) ' . v "I .!.•/• y 197 Cty TNHH MTV DVVH Khang Vi?t Chuv6n dg BDHSG Join g\i tr| Idn nha't va gia trj nhi nhSt - Phan Huy KhSi Hudng ddn gidi x-'+y^ Ddn ^o; max P = —; min P = I ' 4 4 Xem 151 glal (bkng phifdng phap liTdng gidc hoa) trong bal toan 1, § 1, chiftJng xy - iL ie uO nT hi Da iH oc 01 / H(,..,, ' ' - ' •f'(t) = 1-y f'(t) ;I;:/1(5J. ' 1-Z s/ up /g Xem Idi giai trong phan nhan xet cua bai 8, muc 1.2, § 1, chiTdng 2 cua cuon ok , 0 y 6-Vi2l . • £V.v•'ii;;^;i = 4 + 2^/3. Bai 12: Cho x ' + y^ = 1; u ' + v^ = 1. , (., ' ' ' ' ' Tim gia trj Idn nha't va nho nhat cua bleu thiJc: P = x(u + v) + y(u - v). ww w. + trong bai 9, muc 1.2, § 1, chiTdng 2 cuo'n sach nay. . HUdng ddn giai Do x^ + y^ = 1 => X = sina, y = cosa, vdl a e [0; 2n]. Do u^ + v^ = 1 => u = cosp, y = sinp, vdl p 6 [0; 2Tt]. ?! Vi the P = slna(cosp + slnp) + cosa(cosP - sinP) j- + — • Do X > 0, y > 0 va X + y = 1, nen dat X = sin^a, y = cos^a vdi 0 < a < | . ' ; 2. Xem each giai bal toan tren bang phufdng phap suf dung ba't d i n g thufc Cosi HUdngddngidi Luc nay: + thien ham so de glai bai toan. sach nay (dung phiTdng phap li/dng giac hoa). Tim gia tri nho nhat cua bleu thiJc: P = — 0 3 Nhan xet: Xem Idi giai trong phan nhan xet cua bai 5, muc 1.2, § 1, chifdng 2 cua cuon Bai l l : C h o x > 0 , y > O v a x + y = 1. 6 + >yi2 Vl + z^ bo ce , . HUdngddngidi Dap so: maxP = —. Vl + y ' fa ' :li;n1iirii-0. , VlTx^ ^ 1. Ta da phoi hdp phifdng phap "liTdng giac hoa" va phiTdng phap chieu bien Bai 10: Cho x, y, z la cac so thifc diTdng thoa man dleu kien: x + y + z = xyz. , x y z Tim gia tri Idn nhat cua bieu thiJc: P = , + , + / r' • , 6-Vl2 .c om sach nay (dung phiTdng phap liTdng giac hoa). • f(t) y y y y y Vay min P = mlnf(t) = f () Q 3 HU&ng ddn giai ^ 8(-3t^ + 1 2 t - 8 ) t Ta 1-X ^ + - - ^ ^ ^ ~ ^ ] vdl 0 < t < 1 t 4t-3t^ Vay CO bang ble'n thien sau: Bai 9: Cho x, y, z la ba so thufc dufdng thoa man dleu kien: xy + yz + zx = 1. , X y z Tim gia tri nho nha't cua bleu thufc: P = j+ j+ j• 2 4-3t (4t-3t)^ min f(x) = 0 ; -l 0 < 2 a < 7 i = > 0 < t < l . Tim gla tri \6n nha't va nho nhat cua ham so: Dap so: max f(\) = yl2; 1 1 . -sln22a 4 l-^sln22a 4 Bai 8: (De thi tuyi'n sink Dai hoc, Cao ddn)^ khdi D) ' i sln^ a cos^ a 1 I cuon sach nay. f(x) =-^iL trendoan[-l;2]. Vx^ +1 sin^a + cos^ a = sinacosp - sinPcosa + cosacosP + slnaslnp ,| I = sin(a - P) + cos(a - P) = v^cos Tir (1) suy ra: max P = N/2 va min P = - 72 . (1) '' 199 Chuyen BDHSG Toan g\& trj Ifln nhft vk gii trj nh6 nhit - Phan Huy KhSi Cty TNHH MTV DWH Khang Vi«t Nhan xet: Ta c6 the suf dung phiTdng phap dung bat d i n g thiJc Bunhiacopskj v de giai nhuTsau: ^^f- ^'^"f each giai " p h i lUcfng giac h o a " sau day: / T a c o : x ' + y ' = (x + y ) ( x ' * - x V + x ^ y ^ - x y ' ' + y ' ' ) Theo bat dang thiJc Bunhiacopski, ta c6: h a y P ' < 2 ( x 2 + y ^ ) ( u 2 + v ^ ) - 2 (Do x^ + y ' = + TClfd6tac6: -N/2 < P < V 2 . + >/2 2 = 1). . hay tU tint xem khi nao P dat ^id trf Idn nhdt (nhd ^ (x + y ) ^ - ( x ^ + y ^ ) 16z nhdt)? B a i 13: Cho x > 0 , y > O v a x + y = l . T i m gia t r i nho nhat ciaa bieu thvJc: (x + y ) ^ - l z^-1 (z^-l) =>x^y^ = Ta '1 s/ ' up ^^^^^^^^^^^^ ro Ddp so: m i n P = 72 . V i x ' + y^ = 1 /g ; -20Z 2z'-- Do z y^cos ' 5 a Ttf (3) suy ra: max P = V2 ; m i n P = - A/2 . 4 (3) ''ay max P = 72 — ; m i n P = - A/2 . Ban doc hay tifdanh gia ve Unh hieu qua cua hai phiTdng phap "liTdng giac hoa' ^ va "phi lu-ctng giac h o a " trong bai toan noi tren!. Ban thich each giai nao?. 15: Cho ba so thifc x, y, z thoa man he thtfc: xyz + x + z = y. T i m gia tri idn nha't cua bieu thtfc: P = — x"+l —+ ~ . y^ + 1 z ^ + 1 Hu^ng ddn giai T a c 6 : x z + - + - = 1. y y 200 (**) <1 )e ihay da'u b^ng trong (***) xay ra o = (3sina - 4sin^a)( 1 - 2sin^a) + 2sinacosa(4cos''a - 3cosa) ' ' (*) (***) Do x^ + y^ = 1, nen dat x = sina, y = cosa v d i a e [0; 2n]. ' + 5z 5\ .c B a i 14: Cho x^ + y^ = 1. T i m gia t r i Idn nhat va nho nhat cua bieu thiJc: " 2 J Thay (**) vao (*), va c6: |P| < V2 om bai 10, muc 1.2, § 1, chiTdng 2 cua cuon sdch nay. •' -,2 x ' + y2 + 2xy < 2(x2 +y2) = 2 =^|z|<^^. LcJi giai b^ng phiTcfng phap "lifcfng giac h o a " x e m trong phan nhan xet cua Ta c6: sinSa = sin3acos2a + sin2acos3a (x + y r - 1 2'! 4 -47/ + 10z^-5 P = HUdng ddn giai > T a c o : |P| = |l6(x5+ y 5 ) - 2 0 ( x - V y - ^ ) + 5(x + y)| va dat x + y = z, ta c6: Tijf do ta c6: max P= s f l ; m i n P = - y I l P = 16(x^ + y^ ) - 2 0 ( x U + y^) A p dung cong thiJc: xy Hitdng dan giai -xy(x2 x ' + y^ = ( x + y ) ( x ' - x y + y2) = (x + y ) ( l _ x y ) . " P = - ^ c h i n g han k h i X = %/2 ; y = N/2 ; u = v = - C(if +y^f-xY = (X + y ) [ l - x y - x ^ y - ] , . = 1). : C6 the tha'y P = - J l chang han k h i x = y = u = v = (Ldc do ro rang: x^ + y^ = = ( X + y)[{x' )[(u + v)^ + (u - v)^ ] , iL ie uO nT hi Da iH oc 01 / [x(u + V) + y(u - v ) f < (x^ + (I) 201 Chuyen d6 BDHSG Toan g\i t r j lan nh^t IF g\A t r j nh6 n h f t - Phan Huy KhSi Do X , y, z la cac so diTdng, n e n d a t x = t a n - ; - - t a n ^ ; z = t a n J , d day a . p . y e V 2 y 2 2 Cty TNHH MTV D W H T. 10 1 Tiif do suy ra: max P = — <=> x = -j= ; y = V ^ ; z = - i = . 2 2V2 Khid6(l)c6dang: tan^tan^ + tan|tan^ +tan|tan^ = i ffh4n x^*' (*) P = ( l + x2)(l + y2)(l + z2) + ( l - x 2 ) ( l - y 2 ) ( l _ z 2 ) 2 2 2 HUdng ddn iL ie uO nT hi Da iH oc 01 / Taco: 1 1 + x^ = 1 + t a n ^ ^ = 1 +y^=: 1 (1 + c o s a ) ( l + c o s P ) ( l + cosy) > 1 + c o s a c o s P c o s y . iDi/a v a o c o n g thtfc luung g i a c , ta c6: = 1 + tan^ r = cos^-^ 2 Ta 2 ^ s/ 2 1 a - P \ 10 = 1 sin—= - . . 2Y I 2 3 1 9 ro l + y^ (l-x^)(l-y^)(l-z^) 1 + z^ 8 > (1 + x^ ) ( l + y 2 ) { l + z^) + (1 - x^ ) ( l - y^ ) ( l - 7 ? ) tan^ I+2 1+ 1-x^ l + x^ 1+ 1+ 1-Z X ( l + x=)(l + y ^ ) ( l + z ^ ) ' = y = z = 1. Tuy nhien sur dung (*) khong phai la dieu mk ai cung tha'y difdc, trong k h i sur dung (1) va cong thiirc lifdng gidc thi Idi giai tiT nhien h d n ! ^ai 17. Cho x, y , z la ba so difdng thoa man dieu kien xy + yz + zx = 1. T i m gia t r i Idn nha't cua bieu thuTc P = 1+x^ 909 Hiidng + ;^^'"f ~ ^^^"^^^"1^72 y = V2; z = a 2"^ 1 + z^ , V a y dau b^ng xay ra k h i va chl k h i x = f Di nhien cac ban c6 the l ^ m nhuf sau: Tir (*) suy ra max P = 8 o 1 1 , I 7 -Y = -1= > t a n Y - =—7==>z = — 2 8 2 2>^ 2V2 tan|tanl -1 = 0 «tan^ | hay P < 8. • 9 Tiif (*) v d i a = P suy ra (l + x2)(l + y2)(l + z2) V i the hien nhien ta c6: w. 2 . y 1 Da'u b i n g xay ra <=> 1 + x^ 1+ D o x . y . z e [0; l ] = > l - x ^ > 0 ; l - y ^ > 0 ; l - z ^ > 0 . 2 fa cos 9 ce , --cos ) Binh luqn: bo 3 ^ i c o s ^ ^ - 3 sin-—cos——2 3 2J a - P 2 /g 2 3 2a-P om sin-—sin-cos-—— 2 1 o t - P 1+ Vay max P = 8 o x = y = z = 1. .c . y \ 1. Y ww = . y o 1-x 2 A ' I Da'u bkng trong (2) xay ra o da'u b^ng trong (1) xay ra<::>x = y = z = l . ok 2 . y a - P = 3 - 3 sin - — s i n - c o s — — = 3 - 3 2 3 2 2 2Y up = 2 c o s ^ c o s ^ . 3cos^ I = - 3 s i n ^ I + 2 s i n I c o s ^ + 3 r. o 1+ (1) / = 2 c o s 2 - - 2 s i n 2 ^ + 3 c o s 2 ^ - l + c o s a - ( l - c o s p ) + 3cos2^ VayP ; I 03*0 bkng trong (1) x a y r a o c o s a = c o s P = cosy = 0<=>x = y = z = l . 2 1+ 2 Ta c6: c o s a > 0; c o s P > 0; cosy > 0, v i the h i e n n h i e n suy r a : ^ 1 P + cor^ = gidi D a t x = t a n Y ; y = tan^; z = t a n ^ . D o x, y, z e [ 0 ; I ] =>a, p, y e 0 ; ^ cos 2 Phi'f^ng p h a p " l i T O n g g i a c h o a " to ro h i e u qua tren b a i toan n a y ! pai 16: C h o x, y , z e [ 0 ; 1 ]. T i m gia tri Idn nha't c u a b i e u thiJc: . a P Y 71 do - + - + 4- = - 2 Khang Vigt . ' ddn Vi+y^ ViT7 gidi A B C BSt X = tan — , y = tan — , z = t a n — v d i A , B, C e 2 2 2 ^' ^ ^' A B B C C A Tfirxy + yz + zx = 1 => t a n — t a n — + tan—tan — + t a n — t a n — = 1 2 2 2 2 2 2 gii tr| nh6 nhSt - Phan Huy Khii Af B tan— tan — + tan — 2I A 2 Cty TNHH MTV DWH Khang Vi^t B C 1-tan—tan— 2 2 2) ° , B ' C , V tan — + tan— A fB (t\ tan — = cot —+ — 2 2) 2 — tan 2 I2 — + — 12 2. DS'u b^ng trong (4) xay ra o Tom l a i maxP = - Ta s/ z . C = = sm —. 1+z 2 ro /g om .c chi k h i A = B = C = 6O". ok ce fa w. 73 ' ' (4) (do xy + yz + zx = 1) x = y = z = 1. Ta thu l a i ket qua tren! orfr. ' " X ^xy + yz + zx + x^ 7(x + y)(x + z) X x+z Tir gia thiet x + y + z = xyz, ta c6 tanA + tanB + tanC = tanAtanBtanC + C)=> A + B + C = I8O". ,j Vr. t a n B + tanC ^ I-tanBtanC = 2 ^ ^ " ^ f = tan^ A.cos^ A = sin^ A 1 + tan^ A ' 2 TiTdng tir = sin^ B; • 1 + y^ = sin^ C . V a y P = sin^A + sin'B + sin^C 1 + z^ Ta biet r^ng trong m o t tam gidc thl sin^A + sin^B + sin^C < ^ . V x + y Vx + y X <1 ^^l + x2 z^ j + -^—j + 1 + x^ 1 + y'' i+z Do X > 0, y > 0, z > 0, nen dat x = tanA, y = tanB, z = tanC,vdi A , B , C e 1 + x^ X x^ Hudng dan giai Ta c6 DiTa v ^ o xy + yz + zx = 1, ta c6 x+y < Vay CO the coi A , B , C la ba g6c cua mot tam giac A B C . xet: Ta c6 the g i a i b ^ i todn tren b^ng each "phi lifdng giac h o a " nhiTsau: D a u bkng trong (1) x a y ra <:> z = x; x = y. dong thcfi c6 da'u b^ng trong (1) (2) (3) Tim gid t r i Idn nhat cua bieu thtfc P = tanA = - ww o x s y =z = Tijf do theo ba't d i n g thuTc Cosi, ta c6 (3) z+y tanA( 1 - tanBtanC) = - ( t a n B + tanC) bo o A = B = C = 60" A B C 73 <=> t a n — = t a n — = t a n — = — 2 2 2 3 Vl + x^ o up 2 A B C 3 s i n — + s i n — + s i n — < — va dau b^ng xay ra k h i 2 2 2 2 ^ ' ' z+x Bki 18. Cho X, y, z la ba so thiTc diTcJng va thoa man dieu k i e n x + y + z = xyz. A B C NhiTvay P = s i n y + s i n — + s i n Y . NhiTda bict trong m o i tam giac A B C ta c6 X z + Ban thich giai each nao? COS NHn y + x; <^x = y = z=— Jl + tan2 Tird6suyramaxP= I (2) .y + z Congtiirngve'(l)(2)(3)vac6P< | . ^^"i , - Da'u b i n g trong (2) (3) tiT^ng uTng xay ra o V a y A , B , C la ba g6c cua mot tam giac A B C . . A tan A A . A Ta CO ^ = tan—cos— = s m — . 1 2 2 2 Ti/dng tiT CO < 7iT7 2 C 2 B = sin — ; 1 = i+r A + ^ + C ^ ^ y O =>A + B + C=180". /; v v ^ "2 " 2 y - z 2 ^ Ti/cfng t y ta CO 1 - tan —tan — B " y ; ;i ^ iL ie uO nT hi Da iH oc 01 / Chuyen dj BDHSG Toan g\i tr| I6n nhft 2 +y l,x o y = z. Dau b^ng trong (1) xay ra o X +x + z; (1) !_ V a y maxP = ^ A = B = C = 60" o A = B = C = 6 0 ' ' o tanA = tanB = tanC = 73 c^x = y = z=73. _.^„ 2J _Cty TNHH MTV DWH Khang Vi? ChuySn dg BDHSG Toan g\& trj Ifln nhS't va gia trj nh6 nhjt - Phan Huy Khai %\. (De thi tuyen sink Dai hoc. Cao ddn^ khoi D - 2010) PHIfONGPHtPCHlfUBlfNTHItNHJlMSdf TiM GlU TR| itN NHlt YA NHA NHiTr COA HAM Sdf GAwcfn^4. Tim gia trj nho nha't cua ham so: y = V-x^ + 4 x + 2 i -V-x^ + 3 x + l() tren mien xac dinh cua no. HiAhig ddn gidi (Lcfi g i a i v ^ n tat, I d i giai chi tiet xem bai 4, §2 chiTdng 1 cuon sach nay) thong dung nhat de t l m gia t r i Idn nhat va nho nha't cua ham so'. M i e n xac djnh cua ham so la: - 2 < x < 5 iL ie uO nT hi Da iH oc 01 / Cung v d i phiTctng phap bat d i n g thiJc, day la m o t trong hai phiTdng phap B k n g each x e t chieu bien thien h a m so (ma thong thu'cfng ngi/cti ta hay si} • y• - dung phep tinh dao ham), sau do so sanh gia tri ham so' tai c i c d i e m dSc biet 2 V - x ^ + 4 x + 21.V-x^ + 3x - 1 0 (thong thifdng 1^ cac d i e m cifc dai, ciTc tieu, c^c d i e m dat biet n h i / cAc dau Tif do cd bang bien thien sau: mut cua cac doan th^ng xac dinh nen m i e n xac dinh cua h a m so dang xet, nha't, nho nhat phai t i m . y' up ro d\ia Mm so can khao sat /g la thiTc hien mot phifdng phap doi b i e n ddn giSn 0 / 1 Nhiyng bai toan nay thufdng c6 dang ddn gian hoac la trifc tiep khao sAt chieu bien thien h ^ m so' can t i m gid tri Idn nha't, nh6 nha't cho trong dau b ^ i , hoac - Vay m i n y = y s/ Ta y y^ 3 y y y y y y §1. SUr D M N G TRl/C TIEP C H I E U B I E N THIEN HAM SO i I X cac d i e m khong ton tai dao ham...). TiJf ph6p so sdnh ay suy ra cic gia tri Idn D E TlM G I A TR! LdN NHAT. NH6 NHAT ~ > ^ - x ^ + 3 x ^ - ( 3 - 2 x ) V - x ' + 4x + 21 I + /.ui^ ,, , 5 ) y y y y y y y t . \ 3; Bki 3. (Di thi tuyen sinh Dai hoc, Cao dang khoi B) Ttm gid trj Idn nha't va nho nha't ciia ham so: f ( x ) = ve dang dcJn gian v i thuan I d i hdn cho viec t l m gia t r i I d n nha't, nho nhat i;e-^ HUdng ddn gidi Khao sat trifc tiep f (x) va suy ra max f(x) = - i - ; m i n f ( x ) = 0 . l/2 . /p X y Bai 6. Cho X, y e [1; 2]. Tim gia tri Idn nhat ciia bieu thufe: ? = - + HUdngdangi&i D a t t = - . D o x , y e [1;2] = > ^ < - < 2 = > ^ < t < 2 , „ • y 2 y ^ Luc nay ta eo: max P = max f(t), vdi f(t) = t + |-. Ta CO- r(t) = 1 - — = ^ "'t= — o x = + — , 3 3 max f(x) = 4 o t = 0<:>x = 0. A'Aa/i x^/; Bai -1y = 1 - x,vayP = 3"' + 3 ' " ' = 3^" + — . 3" Dat t = m3'.a Do 1 < t =< minF(t) 3. Taco: x P m0 aminP (1) i f(t) = ^ - ^ v d i 0 < t < 2 +t 4 HUdngddngiai +X + 2 vdi 0 < X< ro , _x^+x + 2 1. ce , -x^ x+1 X - X + 1 () : ;0' 1 1 2 y y y y 0 1 . ^^ '^ 3 / /" + ^ ^\ Va y max P = max f ( I ) = f(0) = 1; min P = min f ( t ) = f 0x = y = - . x +y = 1 Ta thu lai ke't qua tren. ^6 bang bien thien sau: + X + 2)' 0 X ww 6x-3 R6 r^ng r ( x ) = -———-—2" w. T a c o : m a x P = 2 max ax:f ( x ) ; m i n P = 2 m i n f ( x ) . (1) () y = l - x , v a y P = /g y+1 x + 1 0 up ^'* om T i m gia t r i Idn nha't va nho nha't cua bieu thuTc: P = s/ B ^ i 9. Cho X, y Ik cac so thifc khong am thoa man x + y = 1. ' va c6 bang bie'n thien sau f'(t) = (2 + 1)' Ta y =- + jlog32 \ V ; V - v , — - > .j(jiO t^j|V /.^'oaxm;; * r 4 X 6 t h a m s^: X = ^ l o g 3 | = j ( l - l o g 3 2) 2''+xy' -^^ 4 9 3 1. Ta con c6 the giai nhiT sau (cung bang phifdng phap chieu bie'n thien ham so) = 33 V2 N l x = — ; y = —, max P = 1 o x = l;y = 0 3 2 2 Nhir vay: max P = max F(t) = m a x { F ( l ) ; F ( 3 ) } - max (4; 1 0 } = 10, , = 2.1 = ^ , 1 1 m a x P = 2 m a x { f ( 0 ) ; f ( l ) j = 2 m a x \ - ' - - } = 1- F(t) . 2J ^- E>e tim gid tri Idn nha't va nho nhal cua ham so: f(t) = ^ - ^ ^ vdi 0 < t < - , 2 +t 4 ta CO the suf dung phu'Ong^phdp bat d i n g thtfc nhiT sau: Ta c6: f(t) = l Do 00=>f(t) t = 0 o CO x = l ; y = () xy = 0 x = 0; y = l . 0 - <=> t = - . D o do: m i n P = - o 3 4 3 1 t=4 ^ 1 xy = — o 1 o x = y = -, 2 (, l ' ( x ) < 27 Ta s/ max l"^(x), up (1) ro ()<\ /g (2) om min l ' ' ( x ) . .c () 1 - cos X = <=> X tos^ X o . 3 . 3 (3) K 3 COS"^ X 3 + 3 27 256 * ^ ,( 2 3 73 , . cos X = — <=> COS X - — (do cosx > 0) 71 6 Vay m a x ? = ^/maxf^(x) 16 <=> x = —. 6 Matkhacdo 0 < x < - = > P > ( ) . 2 Dc thay P = 0 o sinx = 0 cosx = 0 x=0 n X = —. 2 x = () Tit do CO ngay minP = 0 o 71 X = — ;;;.tit>.i;,:,s 2 Ta ihu lai ke't qua trC-n!. f(x) = 7 l + sinx + 7 l + cosx, x e R . Hitdng dan giai 0 maxF(t)= = n »i 11. T i m giti tri Kin nha't va nho nha'l ciia ham so: 0 Vay cos X ^n(t Dau bang trong (4) xay ra 256 , \ w. 2 <=> bo , , max r ^ ( x ) = m a x F ( t ) ; m i n f^(x) = m a x F ( t ) . () 0 V x € 0 ; ^ , nen ta c6: 2 ^ 7t o Ta c6: f'(x) = s i n ' x . c o s ^ J D a t t = c o s ' x = > ( ) < t < 1. 73 1t4n xet: Ta c6 the sit dung bat dang thtfc Cosi de giai b ^ i toan tren nhur sau: Hvldng ddn gidi () 0 < 2 + t < - = > >-. 4 4 2+t 3 V m i n f ( x ) = ^ m i n F ( t ) = 7 m i n { F ( 0 ) ; F ( l ) } = ^ n { 0 ; 0} = 0 . x + y =1 iL ie uO nT hi Da iH oc 01 / Lai Vi5t [rT — ^2^^ 3N/3 =—- Do f(x) > 0 V x maxf(x)= X£R G R , nen ta c6: /maxf'^(x); m i n f ( x ) = Jminl'^Cx) xeR V xeK V xeR I Chuyen dg BDHSG Toan gia tri Idri nha't CtyTNHH MTV DWH Khang Vi^t g\i trj nh6 nhaft - Phan Huy Kh^i HUi'fng dan giai Taco: f^(x) = 2 + (sinx + cosx) + 2^1 + (sinx + cosx) + sinxcosx . Dufa viio cong thijfc: sinxcosx = (-V2y2)} = max{4 - 2 ^ 2 ; 4 + 2>^} = 4 + 2>/2 . om /g max o I = smx + COSX = O COS X =1 .4V ok 71 .c Tiif(l),(2),(3)suyra: maxl f(x) = sin 2x 4x - + cos - + I , v('Ji x e \ 1 + x1 + x' up min s/ Ta 4+ Vay 4; fa 4, X xeR — ww 4j w. minf(x) = 1 <::> I = sinx + cosx = - 1 o cos 7^ x=^7i + o HU(jfiig ddn giai 4x Ta c6: cos i + x^ = l - 2 s i n ^ - ^ . D a t t = sin — 1 + x^ 1 + x-^ Do - — < - ! < < 1 < — va ham y = simx dong bie'n trcn 2 1+x 2 L 71 71 2'2J Luc do: s i n - ^ ^ + c o s - ^ ^ + 1 = - 2 t ^ + 1 + 2 . J + x^ 1 + x^ ce o x = - + k27i, k e Z ftvfti ot.s -sinl < t < sinl bo xeR ,• minRx) = -<=> x = ± 1 . •xiu^mmi xeR 2 Chii y: Bai nay it nha't c6 den 4 Rli giai khac nhau (Xem bai 5, chiTdng 3 (i+N^)t+2+>y2 F-(t) "jufti; +00 1 0 ^ td F(t) 0 iL ie uO nT hi Da iH oc 01 / ItkTI -//.-^vv'^. 4x(x^-l) — , luf do suy ra bang bicn thien ( l + x^) Taco: l"'(x) = Luc do: maxl (x) = maxF(t); minf (x) = min F(t), cl day xeR giai chi tic't trong bai 5, nhan xet 2 cua --r k2n Vay m a x f ( x ) = max F(t); m i n f ( x ) = min F ( t ) , ( l ) xeR x - - - + k27i,keZ 2 A^M/i Af^^- Lam li/dng tif nhuf Iren, cac ban hay giai bai loan sau: |l| - Xem thêm -