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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 = + ^^[-z' -3 • , ,, , TiJf suy P < z-6] Z =X = i ( z ^ - z + 18) = i [ ( z - z + 2) + 16] z=y = i [ ( z - l ) ' ( z + 2) + 16] Do < z < 1, nen tir (3) suy VP (2) > => P > X - y- Z- 1.,.', Sau 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 (1) xay o o X =y=z=1 x^-z2 = Neu X > z > y > Luc ta c6 y(z - x)(z - y) < o xy' + yz' + zx^ < zx' + zy' + xyz o xy^ + yz' + zx^ - xyz < z(x^ + y^) o P < z(3 - z^) o P < - z ' + 3z - + o P < - ( z - l ) ' ( z + 2) + (1) Tff gia thiet x + y + z + xyz = va ket hcJp vdi (1) suy 3z + 7} < < 3x + x^ Tir z' + 3z - < o (z - )(z^ + z + 4) < o z < I • Tirdngtirx^ + x - > o x > (.< Vay ta CO X > , >, ^,.„ , va z < , , N e u x > y > > z Tff bo l ) % + 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 > Tif ta c6: x(y - x)(y - z) < Tilfdo suyra P < Ta s/ up vong quanh, nen chi c6 Ihc gia suf x = max{x, y, / ) hay P < - y ' + 3y - + h a y P < - ( y - z = Hudng ddn giai Hi/(htg ddn giai day giu'a cac bien x, y, z khong c6 linh binh di^ng nhufng c6 vai tro hoan vi • ^ •^i-U% 27xy , d6 < xy < V i X > y > => X - > 0; y - > 0, nen theo ba't ding thffc Cosi, ta c6 x + y - = ( x - l ) + ( y - 1) > 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 ) > ( - x - y ) ( x + y - ) => X + y - xy Tff (2) (3) suy 4-x -y •(x + y - 1) xy + 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 N c u X > x = y = z= 1 > y > z K h i ta c6 (x - l ) ( y - l ) ( z - 1) > => xyz - (xy + yz + zx) + X Cty TNHH MTV DWH Khang Vigt gia tr| nh6 nha't - Phan Huy Khii X Tuf ta C O maxP = + y+ z- 1> mot so biing 1, mot so bang \f2 Theo bat dang ihiJc Cosi, ta c6 = x + y + z + xyz > 4:^xyz(xyz) =>0 X ' + Y " + Z ' + ( X ' Y ' + Y'z' Dau b^ng (5) xay o X^ iTa co: (do x ' + y ' + z ' = 3) Tff (4) (5) suy P < YZ i D a u bang (2) xay 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 ro (3) up (2) Tiir (1) va X > => x(y - x)(y - z) < ^ 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 thi 4X^ nen khong mat tinh tong quat co the gia siJ y la so c( giffa x va z Dau b^ng (3) xay z^+2z + Khi la co X , Y , Z la ba so thiTc khac khong Do vai tro cua cac so x, y, z bieu thufc P co tinh hoan v i vong quanh, P y^ + 2y + 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 ( y - x ) ( y - z ) < x^+2x + 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^ = < w > * * S - : S hoac la ba so x, y, z co mot so bang 0, + y + z - (xy + yz + zx) > - xyz Tir suy P = , r- V2 ; ^'^ VayminP= o x o ddng thdi co dau bang (2) (3) o X = y = z= = Y = : Z < = > x = y = z = , , ^ , 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 va Ihoa m a n d i l u k i e n xyz = \ 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 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 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 Dal X = „ kYZ kZX kXY (ban doc -,z - • tif nghiem lai) tV) -.Ml r r x^ ^ YZ VYZ + 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 D a t X = -n= '- Y = T7= ; Z = -7= hay x = — ; y = — ; z = — 3/7 3/7 3/r ^ 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 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 (2) xay « 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 (3) xay X = Y = Z o X = y = z = '* JJ^u • B&i 36 Cho X, y, z la ba so thiTc difdng va thoa man dieu kien xyz = 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 = lx = y = z = I XYZ Dau bhng (2) xay o X = Y = Z .:,,„,„1, I Dau bang (3) xay o X = Y = Z = o x = y = z = fY X , Luc d6 bieu thtfc P c6 dang P = —+ X X U Y Z z (X + Z - Y ) ( Y + X - Z ) ( Z + Y - X ) Vay ta c6 max? = I o x = y = z = ' ca mau va tuT cua phan so vc' phai dcu diTdng, nen tir (2) suy P > X = — ; y = — ; z = — k h i ta c6 X, Y, Z la ba so thiTc diTdng T i r ( l ) ( ) s u y r a P < +(Z^-XYf | r (XY + YZ + Z X ) ' > bo dat I = X " + Y ^ + Z % X ' Y ' + Y ^ Z ' + Z ' X - - 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 = • Z I Luc bieu thufc P c6 dang P = (trong ba: tap tren da dung phep thay bien vdi k = 2) , Y^ x = TYZ 7;r;y = — ZX; z = - XY Nhifthc'co the vie't lai phep thay bien x ' iL ie uO nT hi Da iH oc 01 / • J ' Khi ta z-1 HU&ng ddn giai D a t X = ^ ; Y = ^ ; Z = ^ T^i + y-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 phan nhan xet cua bai 33 vdi k = 1) Khi bieu thtfc P trd -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 Cho x, y, z la ba so du'dng vii xyz = , , , 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+ (z + 1) • 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 + Ta up T i m ilia t r i Idn nhat cua bieu thu'c: 1 - xy om ok bo + Z(X - Vf > O (6) dan - zx giai Neu + 9xyz - X - y - z < * Neu + 9xyz - x - y - z > Theo bat dang thtfc Cosi, ta c6: x + y + z = (x + y + z) = (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 (7) xay X = Y = Z x = y = z = =>0< + x y z - X - y - z < => (1 + 9xyz - X - y - z) ^ xet: 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 nhan xet cua bai 33) de giai nhieu bai loan t i m gia 1 - xy V a y minP = l o x = y = z = l Nhan - yz =>P 0, nen (6) dung va dau bang (6) xay c > X = Y,= Z T i r ( ) v a ( ) c P > 1 +• Do x^ + y^ + z^ = => |x| < 1, |y| < 1, |z| < => - xy > 0, - yz > 0, - zx > fa Xr w + Y(Z - * ww Zr ce Cothe thayrang(X + Y + Z ) ' > X ' + Y ' + Z ' + 24XYZ •+ /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 = < ô 1% HUdng c VX^ + Y ^ + Z- + 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^ = s/ (3) / X + Y + Zt^ Tir (2) (3) suy P > ' 7(X + Y + Z)(X-^ + Y-^ + Z-^ + X Y Z ) Dap so': minP = ^ .rfn.'/U < V(X + Y + ZHX"* + Y-^ + Z-^ + X Y Z (5) o X ( Y - f ''"X'-it'Oilt,.-:': ^'^y^ + Z + Vx^+21x + That vay bang cac phcp b i c n d d i sd cap, la c6 + - \l4y^~+y^^ p = + X Y Z + V Y VY-^ + X Y Z + N/ZVz'^ + X Y Z Dau bang (4) x i i y T i m gia t r i nho nhat cua bieu thu'c: + Y Z + Y ^ y ^ + 8ZX + zVz^ + X Y Vx-^ V4x^ + X + + 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 = Dau bang (2) xay o X = Y = Z xVx^ iL ie uO nT hi Da iH oc 01 / xVx^ d: Da'u bang (1) xay 1-yz + ' l-zxy < 1-xy + 9xyz - x - y - z = o X + y + z = 9xyz , n > t a c — + — > va da'u dang thu'c xay 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 ^ ^ < i 1-xy Dau bang (3) xay o (4) om c ok bo < -+ 1-yz y^ + x^ z^ + x^ zx z^ •+ • 1-zx 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) up U ro 1^ 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) xy < 2xy (x + yr 1-xy - ( x ^ + y ^ ) < (x^ +z^) + (y^ +z^) (do x^ + y^ + z^ = va (x + y)^ > 4xy) Ap dung ket qua sau day: (9) =1 z , x-y V =z (1) (2) (3) ^ y Lai CO - > , ^ > => - - 1-y > X y y z , X y • (4) =:>- + - < l + - Da'u b^ng (4) xay o x=y o y = z y =l z C6ngt£rngve(3)(4) vkco - + ^ + - + ^ < + -X + -z *i v : • y (5) y z z x Dafu bang (5) xay o dong thcJi c6 da'u b^ng (3) (4) 'x^y _y = z Mat khac tCf gia thiet c6 < x < ; l < z < = > x - z < ; z - x s => (X - 2z)(z - 2x) > , (6) X z oDau2x^bhng + 2z^ ^ 5xy (6)oxay- ra+ - o< -x=:2z z=:2x z X 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 = - + - + - + - + - + - => x^ + y^ + z^ > - ( x y + yz + zx) ;Z= (y + 1) - R6 rang ta c6 (x + y + z)^ > =:> x^ + y^ + (z + 1)^ 1+y >'•' ^ ^ ' ( X + l)2 X + ^ = 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 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) fa x = 2; y = z = w VaymaxQ= — ' """^'h z 1+x Taco ok = z= l;x = (x + 1)^ bo ' o y 45 y DatX= 1::^; Y = ro x " Hiidngddngiai 9 => - ( x + y + z) - (3x + 2y + z) > => 3x + 2y + z < - (x + y + z) 4 , Bai 40 Cho x, y, z la ba so thiTc du'dng va thoa man dieu k i e n xyz = 11; 2] =:> y + 5z > > 3x —+ —+ X y (5) (1)), va diTa vao m o t ba't dang thiJc cho triTctc dc giai b a i toan dat maxR = hay maxP = 10 • (3x + 2y + z) : Vay minP = x, y thoa man (5) T a c o - ( x + y + z ) - ( x + y + z) = - ( y + z - x ) 4 X, ^ pinh ludn: 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 = thoa man (5) (dox>y>z) ^ y = z = l;x = 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 = X iL ie uO nT hi Da iH oc 01 / X = 2z hoSc z = 2x (x + y)- = + xy (3) Dau bhng (4) xay 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 ^ ( ( 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 Trong b a i 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 = T i m gia t r i nho nhat cua bieu thiJc: + 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 ; 1+ z = 1+ Z + (1 + X ) ( l + Y ) ( l + Z)" (6) c N h i / v a y P = - [ ( + X ) ^ + (1 + Y ) ^ + (1 + Zf 4L (4) ce , , w Vay ta CO P > => minP = ' ' Difa vao dong nha't thiJc: 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 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) '^ ^/xyz (x + y ) ( y + z)(z + x ) xyz (1) „ -w* 3^/?77 ' (x + y ) ( y + z)(z + x ) ^ x + y + z ^ ((1) chi^ng m i n h de dang v^ x i n danh cho ban doc) Da'u bang (2) xay ra x = y = z = l 2(x + y + z) 1+^ \ 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 = 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 (2) (3) 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 = • '' 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 = Binh luqn: T i r d o de dang suy (1 + X)^ + (1 + Y)^ + (1 + Z)^ + (1 + X ) ( l + Y ) ( l + Z ) > (ban doc tif chtfng minh l a y ) o M Vay minP = - x = y = z = ( ro ; 1+ y = x +y+z (5) (do x y z = l ) ~ 3' o x - — — ; 3(x + y + z ) - (x + y)(y + z)(z + x ) ^ x +y+z /g + X= > Da'u bang (5) xay X = ^ - ^ - Y ^ ' - ^ ; Z = ' - ^ , 1+x l+y 1+z \ I X = x) > ^ / x V ? • Ta (1 _^p^ •); » s/ ( l + y) N om => + xr r +- + y ) ( y + z)(z + ok Dat (1 (x ' o i • (X + y)(y + z)(z + X) > - ^ x ^ y ^ z ^ (x + y + z) X e t bai toan tifcfng tif sau: ^ + x = y = z = iL ie uO nT hi Da iH oc 01 / A^/ia/i xet: (3) i Oa'u bang (3) xay o „ P= x +y+z L a i theo ba't d i n g thuTc Cosi, ta c6 ^/xyz < 2x +y+z^^x +y +z (x + y)(y + z)(z + x ) — xyz t xyz xyz 2(x + y + z) ^ x + y + z ^ _ (2) L a i CO ^ t J ^ ^ > (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 = X = y = z > TiMGlATRIltfNNHKtlNi NHiNHlttCdAHAMStf B a i 42 Cho x, y, / > va ihoa man xy/ - 1 T\m gia trj nho nhat cua P = (x + y)(y + /.)(/ + x) - 2(x + y + /) fx + y ly + z Iz + x 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 nhi^ng phrfdng phdp hay suT dung de t i m gia t r i '' Idn nhaft, b6 nhaft cua h a m so' 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 - ( x + y + z ) tKt" (X T h c o ba't dang thiirc Cosi, ta c6 x + y + z > ^/xyz L a i CO xy + yz + zx > ^ / x V z ^ = (do x^yV^ TCr ( I ) (2) (3) suy P > d) ,.>,Vr* - + y + z) - > - = 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$c la dieu k i e n bai toan ban dau c6 dang: x^ + y^ = a^ a > 0, - HoSc la cdc bieu thtfc da cho ban dau g^n lien v d i m o t h? thtfc liTdng gidc quen biet nao T i m gia t r i Idn nha't cua bieu thiJc P = ^ x y z + J ( l - x ) ( l - y ) ( l - z ) Htidng ddn giai Do x, y , z G [ ; ] , nen dat x = sin^A, y = sin^B, z = sin^C, zx) r—^ ' («) Tilf (7) (8) suy (x + y + z)(xy + yz + zx) > x y + yz + zx + x + y + z + K h i d < s i n A < l ; < s i n B < l , < s i n C < l ; < c o s A < 1; < c o s B < l , < c o s C < ' ' r'I iv Ta c6: cosAcosBcosC xy + yz + zx + x + y + z + \ " o (x + y + z)(xy + yz + zx) - > xy + yz + zx + x + y + bai toan t i m gia t r i Idn nha't, nho nha't c6 the suT dung phtfdng phdp liTdng gidc hoa thi/dng c6 cac dau hieu de nhan bie't sau day: 3(x + y + z) - - 2(x + y + z) That vay dufa vao (*) suy = Cic (3) 1) TrU'dc he't ta chii'ng minh rhng o(x tim gia t r i Idn nhat, nho nhat da cho ban dau (2) = (do xyz = ) De thay dau bang (4) xay x = y = z = (x + difa vao phep tinh Itfctng giac ta se de dang hdn trong viec g i a i b a i toan s/ => P > (X tant, ) ta diTa bieu thiJc va dieu k i e n cua bai todn ve dang luTcJng gidc Tir d6 o rx = l (*) LLy = l O) Chuy6n gj BDHSG Toan glA trj Idn nhat va g i i tri nh6 nhS't - Phan Huy Khii sinC = l Dau b^ng (3) xay sinAsinB = Cty TNHH MTV DWH Khang Vi$t => tan^a tan^p tan'y tan^ > 81 => x''y''zV > 81 => P = xyzt > 'z = l x=0 Dau bSng (7) xay dong thcJi c6 dau b^ng (4), (5), (6) (**) a = P = Y = y =o Tir (1), (2), (3) c6: P < cosAcosB + sinAsinB =>P< cos(A - B) ^ V i cos(A - B) < va dau b^ng xay va chi A = B, nen ta c6: ^hdn xet: Hoan toan tiTcfng tif, ta co ke't quS sau: C h o x > , y > , z>Ovlk —^—- + —J—+ — + + x^ + 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^ , (5) sin^ > 3^/cos^ a cos^ pcos^ y (6) sin^asin^PsinVin^ > Icos^a cos^p cos^os^ - 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 bai toan 1, bai 1, chiTcfng cuon sach n^y w ww Tir (1) suy ra: sin^a = cos^p + cos^y + cos^6 Xem IcJi giai bai toan 3, § 1, chiTdng cuo'n sach Ddp so: max f(x) = 2>j2 ; f(x) = - ce cos^ (; — ^ = 1 + t^ Hudngddngidi VI vay dieu kien: + -r + j + j =1 • + x^ + y^ l + z^ + Lap luan tiWng tif, ta c6: , xdc dinh cua no .c I + y" = — - • cos'p /g Dat x^ = tana; y^ = tanP; z^ = tan y; t' = tan vdi a, P, y, e ' Ap dung bat dang thiircCosi, ta c6: , Hitdng din giai Dap so: max P = 3; P = - Hudngddngidi o cos^a + cos^p + c o s \ cos^ = 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 = ^ ( ^ + x y ) + 2xy + y^ Bai 2: Cho x, y, z, t > va thoa man dieu kien: 1 1 T + T + + =1 • CQS Qt ; Bai 3: (De thi tuyen sinh Dai hoc Cao dann khdi B) Tir ta CO max P = o x, y thoa man (6) Tilfd6: + x" = + tan^a = - — ; - ; Khi neu P = xyz, thi P = 72 x=y=z=l -'^ (8) < ' , iL ie uO nT hi Da iH oc 01 / Dau b^ng (4) xay o dong th5i c6 dau b^ng (2), (3) z - ; x = 0;y = ' « x = y = z = t= (4) Vay P = o x, y, z, t thoa man (8) P^tx= tancp vdi x e "2' Khi 66 ta c6: + 4x^+3x^ + 4tan^(p + 3tan'*(p , , ^ \ — "2— = 5—21=:(3 + 4tan''(p + 3tan^(pjcos ( l + x^) ( l + tan2(p) i - - s i n ^ ( p + sin^2(p ' • = 3cos'*(p + 4sin^(pcos^(p + 3sin'*(p = \ ' = 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) = - ^sin^ 2(p, vdi cp e Tir n 7C do: f'(x) = (8x + 12x'')(l + x ^ ) ^ - x ( l + x^)(3 + 4x2+3x'*) (l + x '2''2) ^(8x + 12x-'')(l + x ^ ) - x ( + 4x^+3x^) Ta tha'y ngay: F(cp) = - ^ = | « si 2(p = 1; (l + x^)' Vay C O bang bien thien sau: maxF((p) = - = o s i n ^ ( p - De thay: 2'2, * ' nen tijf bang bien thien tren suy ra: 2x^ Ta 3x'* +4x^ +3 Goi m la gia tri y cua f(x) Khi c6 phufdng trinh sau (an x) + 4x^+3x^ + 2x2+x'* /g Vay maxf(x) = 3x = om xeR bo (4) ww w • Tir va theo (5) suy ra: f(x) = | x = ±1 Ta thu lai ket qua tren 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 = la mot gia tri ciia f(x) Dau bang (4) xay o x^ = o x = ± Taco: f(x) = '"'^ Neu m = 3, (6) co dang: 2x^ = o x = (3) B a y g i t i r ( l ) , ( ) l a c : f(x) > | V x e R ^^i-^'' '> (m - 3)x'' + 2(m - 2)x^ + m - = (6) VxeR fa J Tir theo (2) suy ra: — (X^+1)^ >4X^ (Ivj = m nghiem Do x ' + 2x^ + = (x^ + l)^ >OVx,nen ok c , ±1 Ta c6 each giai khac niJa bang phiTdng phap mien gia tri ham so nhifsau: ro up s/ Viet lai f(x) dirdi dang: f ( x ) - - =3 — r — • x^+2x^ + X +2x''+l Do >0 ly} „ Vx, nen iCf (1) suy ra: f f ( x ) < V x e M lf(0) = x''+2x^+1 xeR + ' maxf(x) = X = 0; minl"(x) = - x xeR xeR thtfc) VXGR ( l + x^) 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 lim '^^^^^ ^'^f ^ maxf(x) = o x = ; minf(x) = ^ o x = ±1 xeR M a t khac: f(x) XER (i + x ^ r •' f'(x) 2'2, '^'^^ 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 Ta c6: 6-x>0 -3 O n e n t^ = + 2V(3 + x ) ( - x ) (2) ** Theo ba't d^ng thtfc Cosi, suy ra: I < t^ < + [(3 + X ) + (6 - x)] => < t^ < 18 => < t < 3^/2 (3) iL ie uO nT hi Da iH oc 01 / DE TlM G I A TR! LON NHAT, NHO NHAT rCf (4), (5) suy ra: max f(x)= max F(t) = F ( ) - , 9-3v^ -34z + N/Z) + ^ - 4x) + ( y ' + ^ - 4y) + ( z ' + ^ - 4z)] - = 27- i (1) 3P = 3(x + y)(y + z)(z + X ) - ( ^ + ^ X-'=\/x o x = ( d o x > ) DSu b i n g (1) xay o / - '! = 4x (1) iL ie uO nT hi Da iH oc 01 / A p d u n g h a n g d i l n g thuTc x^ + y^ + z' = (x + y + zf > 4^x^(^f (4) xSy o x = y = z = , , + 3/^ + ^ ) > 12 j , , y^^j.,, ^ (4) ^ , , I A p dung h^ng d i n g thut ta Viet lai (4) diMi dang sau: (chu y X + y + z = 3) > • Xet ham so f(t) = t^ + ^ - t v d i < t < ' r(t)==3t2+^-4 (x + y + z ) ' - ( x + y)(y + z)(z + x ) + ( ^ + ^ +^ ) > 12 = > - ( x + y)(y + z)(z + x ) + ( ^ + 3/5^ + ^ ) > • ,4 , hay f ' ( u ) = o 3u^ + - - - (d day dat u = /g V t g- om ) u ^, , c 30^ - 4u + = o (u - l)(3u^ + u H 3u - 1) = bo Vay phiTdng tfmh f ' ( u ) = c6 nghiem u = va mot nghiem Uo nen < Uo < ce TiJf suy phiTdng trinh f ' ( t ) = c6 mot nghiem t = va mot nghiem t ^ (x^ + y^ + z \) Da'u bSng (5) xay dong thcJi c6 da'u b^ng (2) (3) (4) F(t) = F(N/2) = + 3V2 o t = >/2 iL ie uO nT hi Da iH oc 01 / ^ ~ < (t-lr ^ x =y=z = — Tir (2) suy F(x) = F(t) = + V < » t = N / o x - - 0 V < X < Khi F(x) la h^m dong bien tren < x < 1, nhif vay vdfi moi < x < tac6F(x)t Do F(0) < 1; F ( l ) < 1, nen suy vdi moi < x < 1, ta luon c6 F(x) < NhU"vay ket hcJp lai, ta luon c6 P < Vay max? = jj,^,^ .^^^^ 11, §2, chiTdng cuon sach phap chieu bien thien ham so de giai bai toan da cho G&m^ up '' • s/ da coi no chi la ham cua mot bien (bien x chang han) Sau sur dung phtfdng Ta Trong bai toan tren mSc dau bieu thffc P phii thupc vao ba bien x y, z nhifng ta ro CAch giai cung c6 the ap dung dc giai dc thi tuyen sinh Cao d^ng dai hoc z z+x bo - xy + y^ ce thoa man dieu kien (x + y)xy = •^:£.mk~'i> > < t } + (|jdi "tot TlM GUI TR| UlN NH/fr NHi NHfr Gift HiUH Stf Gia sur ta phai tim maxf(x) hoac m i n f ( x ) , cf day D la mien xac dinh cua bien so x * " X6D ' • " ' ' Khi de suT dung phiTcfng phap mien gia trj h^m so de giai bai toan tren ta lamnhU'sau: : o :•*••;y;x>z , (?:>sfun6v; Nhgn xet: Xem each giai bai toan tren bang phiTdng phap bat dang thuTc bai Nhqn xet: " i-' teR w Tim gia tri Wn nhat cua bieu ihiJc P = -4- + ' x^ • y2 • ww xeD :W-.::.,::v.- _, HU6ng ddn giai CO nghiem Tuy dang cua (1) ma ta c6 di/cfc dieu kien de (1) c6 nghiem Noi x-* (1) x-y y T t r r i ) v a g i a t h i g t s u y r a P = (-^y)(^^yy^y Chung cac dieu kien co dang: a < m < ^ +y \2 (2) U n^o cua X e D, de c6 da'u bang tiTdng lirng xay d ben phai, ben tnii cua (2) yj Noi each khac ta tim di/dc x,, e D, x, G D cho l"(x„) = P; f ( X | ) = a ,.2,t2 Ket hdp (2), (3) suy ra: maxf(x) = p va minl(x) = a D5t X - ty Khi 66 tit gia thiel ta co (y + ty)y.ty = t y ' - ty + y = y (t - t + 1) xeD => X f +t == ty = t^ - t + t+1 (2) B^ng each giai phiTdng trinh cu the f(x) = P; f(x) = a ta suy difdc vdi gia tri X xy t' - t + P (3) (3) xeD E>ay cung la mot cac phifdng phap hay dung de tim gia trj Idn nhat, nho nha't cua ham so 227 Cty TNHH MTV DWH Khang Vi^t Chuy6n BDHSG ToAn gi^ trj Mn nhait vk g\i trj nh6 nhSt - Phan Huy Kh5i Bai Tim gia trj Idn nhal ^ nho nhat cua ham so: x ^ + x + 23 niaxf(x)= — x = 2; f(x) = - x = - xeR xeR Ta thu lai ket qua Iren Ban ihich each giai nao? ^ x^ +2X + 10 Ta lai c6 the giai bang phi/cfng phap bat dang thtfc nhiT sau: HuAng dan gidi Gpi m m gia Iri y ciia f(x), phiTclng Irinh sau day (an x) f ( , ) ^ ^ t t ^ - ^ ^ i l l _ = + x ^ + x + 10 x ^ + x + 10 (x + l ) ^ + ' x ^ + x + 10 Ta c6: CO nghiem V i x^ + 2x + 10 > (Vx), nen I (l)2x' + 7x + 23 = m(x^ + x + ) (2) =1 ,,, K h i m = - thi(2)c6dang:x^ + 8x +16 = 0x = - bo ce fa w ww — ^ ^ ^ ^ " ^ , va c6 bang bie'n thien sau: ( x + x + 10) -4 Tif suy ra: + +00 (**) ^ o x = -4 Ta s/ ro ok c • Vay maxf(x) = - o x = 2; minf(x) = - o x = - xeR xeR Nhqn xet: Ta CO the giai bai toan trcn bkng phiTdng phap chieu bien thien h^m so nh\S sau -00 X +1 Vay m a x f ( x ) = - o x = va minf(x) = - x = - up chi khi: /g • R6 rang: y' = (X + 1)^ + •Ro r^ng f(x)=: - x = 2, f(x) om K h i m = - t h i ( ) c d a n g : x ^ - x + = c ^ x = 2 • \ -'t»l!b' is • -• (x + 1)^ + Tiif (**) va (*) suy ra, Vx e R ta c6: - < f(x) < 2 Neu m ?t 2, (2) c6 nghiem va chi •1'^^ ' (2m - if - 4(m - 2)(10m - 23) > ,^ o W - m + < o - < m < - v a m?!:2 2 Ket hdp lai, suy (2) c6 nghiem (ttfc \l (1) c6 nghiem) ^5 - • m < — 2 ••,1 x+1 => — /r53^ ^i+>/r53 (1 - 3m)^ o 2m2 - 3m - < c : > - - < m < Ta thu lai ket qua tren 231 Cty TNHH MTV DWH Khang Vi§t BDHSG Toan gia tr| Ifln nh3ft vS gia trj nh6 nhat - Phan Huy KhSi J U j ro ce ww iL ie uO nT hi Da iH oc 01 / 4t 1-t^ • - '< 2sinx + cosx + l _ i + t^ + t^ ^ 4t-f2 sinx-2cosx + 2t ^ - t ^ ^ ^ 5t^+2t + l + t^ + t^ D e n d a y t a c o : maxf(x)=maxF(t); minf(x)= minF(t) xeE teR xeR teR Ta lai c6 the suf dung phiTdng phdp mien gia tri ham so', hoSc chieu bien thien ham so de lim maxF(t), minF(l) Cac ban thuT giai tie'p xem! leR leR minf(x) = - — o x = -—+ k27t, k e Z x€R 2 Nhtf vay ta thu lai ke't qua tren Ro rang phifcfng phap mien gia tri ham so Uti the' ro ret thi du n^y! - o r Ta Bai (De thi tuyen sink Dai hoc, Cao ddn^ khoi B) Cho X va y la hai so thiTc thoa man dieu kien x^ + y^ = Tim gia trj \dn nhat va nho nhat cua bieu thtfc: p = l + 2xy + 2y^ Hudngdhigiai g|j • Xem IcJi giai bang phiftJng phap mien gia tri ham so each giai bai toan 3, § 1, chU'dng cuon sach • Xem IcJi giai ke't hdp phu'dng phap lifcJng giac hoa va mien gia tri ham so each giai bai toan 3, §1, chiTdng cuon sach n^y Bai Cho x, y la cdc so thiTc tiiy y ? , Tim gia tri Mn nhat va nho nhat cua bieu thtfc: P = ^ / ^ ^ ^ ^ x^+y^+7 Hudng ddn giai fa - bo -7\ ]_ Tir (diTa vao linh chu ki 2n cua f(x)) suy ra: max f(x)= o X = k2n, k e Z, xeR U ok -n cosx - sinx - f'(x) f(x) X c Tif ta CO bang bien thien sau: - = 72 sin /g X w Ta c6: cosx - sinx - = -v^ sin om m - n < X < 7t Xet each giai khac nifa sau day: E)att = t a n - , k h i d : up 2sinx + cosx + l Xct phiTdng Irinh: =2 (3) sinx-2cosx + ^ Ta CO (3) o cosx = o x = k2n, k eZ i , ^ 2sinx + cosx + l , " i i i m> TiTdng tuT: — smx = - l o x = — + kin, keK sinx-2cosx + 2 Vay m a x f ( x ) = o x = k27t, k e Z , f(x) X = -— +k27t, k G Z xeR 2 ^ f'(x) = NHnxet: ' ' (sinx-2cosx + 3) X(5t each5 cgiai o s xbkng - c ophiTdng s ^ x - s phdp i n x - 5chieu s i n ^ xbien ^thienc oham s x - sso i n xsau: -1 (4) = 5Ta c6: (sinx-2cosx + 3)'' ( s i n x - c o s x + 3r^2 • (2 cos X - sin coscua x + cosx 3) - (cos x + -2 sin Tir (4) suy da'u cuax)(sin r(x)xla- da'u - sinx x)(2sin x + cosx +1) R6 rang f(x) la ham tuan hoan vdi chu ki 2n, nen ta chi can x6t f(x) vdi s/ Chuyen ii Goi m la gia tri y cua P Khi phu'dng innh sau (an x va y) x + 2y + l CO nghiem Ta c6: ( l ) o m x ^ - X + m y ^ - y + m - = (2) • Neu m = thi (2) c6 dang: x + 2y + = (3) Ro rang (3) c6 nghiem (thi du X = - , y = la nghiem cua (3)) Vay m = la mot gid tri ma bieu thiJc nhan • Ne'um?!;0 Vi (2) c6 nghiem, nen noi rieng: , A = - 4m(my^ - 2y + 7m - 1) > ' ' ^' ' ' ' '' Wy^ - 8my + 28m^ - 4m - < ^ '• (4) Ta sur dung ke't qua sau; Bat phiTdng trinh: at^ + bt + c < (vdi a > 0) Chuyen 6i BDHSG Toan gJA trj Idn nhjt vi gii tr| nh6 nhaft - Phan Huy Khai CO nghiem va chi khi: Cty TNHH MTV DWH Khang Vi? ' I Vay trtfcfng hdp nay, ta c6: < P < - ^ - 4ac > (that vay ncu b^ - 4ac < thi at^ + bt + c > Vt) TCr do (4) CO nghiem nen: f 5'-16m^ -4m^(28m^ - m - l ) > 28m^ - m - < (dom^>0) o — ^ < m < — (va m 7^ 0) •• H : „.-^^.- • IJH thi eh^c ch^n P > -iV-^^; T6m lai ta lu6n lu6n c6: - — < P < - « 14 Dau b^ng ben phai xay x = va y = 2, ok Xet ba kha nang sau: X Nhir vay: max P = - va P = - — Ta thu lai ket qua tren bo o x =—;y = 14 c Ttr ta c6: max P = - o X = 1; y = 2, minP= — ro 28 X = up Ta 14 /g /\ 0: om o x r 14^ ( 1^ 49 r +—y + — = o x +— + y + — 5/ f , f25x2+49 = -70x Dau bang xay •{ [25y^+196 = -140y y =2 X + 2v +1 5_ K h i m = : — ; ^ , t h i ( l ) c dang: — \ x^+y^+7 14 14 +—: 25(x + 2y + l ) , Vay tri/cfng hdp nay, ta c6: •Mr ' y=2 y2+l-4y 25(x2+y2+7) +7-2x-4y-2 = o x = l (25x^ + ) + (25y^ +196) - 70 > (-70x) + (-140y) - 70 - -70(x + 2y +1) > Q o ( x - l ) ' + ( y - 2)^ = l = 2x Mat khac, ta c6: J, Mat khac m = - , thi (1) c6 dang: ^^2y + x^+y^ + V [x2 + Neu X + 2y + < thi ch^c ch^n P < t , , - v r • ^5 iL ie uO nT hi Da iH oc 01 / Ket htJp lai suy ra: • •{ Dau bling xay , „ , K h i d : D - { ( x ; y ) : x ^ + x y + y^