2015 Author : Nguyn Ngc Hng FACEBOOK / toanlihoakhoahoc 6/6/2015 MT S DNG BI TON V BT,CC TR I S V PHNG PHP GII 22 Pages Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 2 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Mt s dang bai toan bdt , cuc tri dai so & phuong phap giai Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 3 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Mt s phng php gii bi tp bt v cc tr i s ---o0o---:Nguyn Ngc Hng:---o0o--- I.Dng 1: Cho cc s thc dng x (i = 1, n ) v A l tng i xng ca cc bin x . Tm GTNN ca biu thc : 1niiixSA==. PP chung : -B1:Do A l tng i xng ca x nn ta d don du bng xy ra khi v ch khi 1 2...nx x x = = = -B2: (Bin i) p dng BT Schwarz : 221111 S..niniinii ii iixxSxAx A===| | |\ .= > = -B3: Chng minh 1 0S S >(0Sl hng s ) ri kt lunDu = 1 2...nx x x = = =. VD1: a,b,c > 0 . Tim GTNN : S=2 2 2a b cChoa b c b c a c a b+ ++ + + + + + Gii: Ta c: ( ) ( )( )2 2 22 2 22 22 2 2 2 S=2 2 23 3 3a b ca ab ac b bc ab c ac bca b c a b ca b c ab bc aca b c ab bc ac+ ++ + + + + +++ + +> =+ + + + ++ + + + + Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 4 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - p dng BT ( )23a b cab bc ac+ ++ + s

ta c : ( )( )( )222343a b cSa b ca b c+ +> =+ ++ + + Du = a=b=c Suy ra min S=3/4 a = b = c > 0 II. Dng 2: Cho phng trnh(x) = u c nghim . Trong (x) l mt a thc . CMR : 21ni oia p=>(opl hng s ) PP chung : -B1: Bin i ( ) 0 ( ) ( ) f x Px Qx = =.Trong : ( ) Pxl mt phn ca ( ) f x c h s l cc hng s; ( ) Qxl mt phn ca ( ) f x c h s l o. -B2: (Bin i) p dng BT Bunhiacopxki : 22 2 211 11( )( ) . ( )( )n ni ii iP xP x a Qx aQx= =s > (1( ) Q x l a thc vi bin x) -B3: Chng minh 201( )( )P xpQx > pcm VD2: Cho phng trnh x4+ox3+bx2+ox +1 = u (1) C nghim CMR: o2+b245 . Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 5 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Gii: Nu x l nghim ca (1) th x u.Ta c: (1)(x4+1) = (x3+x)o +bx2

(x4+1)2 (o2+b2)|(x3+x)2+x4] o2+b2(x4+1)2x6+3x4+x2 =(x2+ 1x2)2x2+ 1x2+3 . t t=x2+1x2 th t 2 Ta c : 22 2( )3ta b f tt+ > =+ Xt t1, t2 bt k e |2; +) (do t 2) sao cho t1 > t2 > u . Ta c: (t1) (t2) = t1t2(t1 t2) +S(t12t22)(t1 +S)(t2 +S)> u Suy ra f(t) ng bin trn |2; +) suy ra (t) (2) =45 => pcm III.Dng 3: Cho A,B,C l tng i xng ca cc bin dng x,y,z .Tm GTNN : S = _xA+_B +_zC . PP chung : -B1:Bin i : tm s k sao cho x y zA kx= = = x y zS kkxA kyB kzC| |= + + | |\ . -B2: (nh gi S) p dng BT Cauchy (AM-GM): Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 6 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2x y zS kkx A ky B kz C| |> + + |+ + +\ . -B3: Tm min ca biu thc A=x y zkx A ky B kz C+ ++ + + ri sau kt lun min S= 2 k.min A. VD3:Cho a,b,c l di ba cnh ca mt tam gic.Tm GTNN : 2 2 2 2 2 2a b cSb c a a c b a b c= + ++ + + Nhn xt : Vi a=b=c th 2b+2c-a=3a nn ta c k=3 Gii:p dng BT AM-GM ta c : 33 (2 2 ) 3b(2 2 ) 3c(2 2 )a b cSa b c a a c b a b c| |= + + | |+ + + \ . 2 33 2 2 3 2 2 3 2 22 32 2 2 2 2 2 2 2 23 3a b ca b c a b a c b c a b ca b ca b c a b c a b ca b ca b c a b c a b c| |> + + |+ + + + + + \ .| |= + + |+ + + + + +\ .| |= + + = |+ + + + + +\ . Du = a=b=c .Vy min S= 3 a=b=c. IV.Dng 4: Cho cc s thc dng o (i = 1, n ) ; A l tng i xng ca cc bin o v ( )1

niia q q const== =. Tm GTLN ca biu thc : 1mmiiS k A== + . Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 7 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - PP chung : -B1:Do A l tng i xng ca o nn ta d don du bng xy ra khi v ch khi :( )1 2...n iqa a a k A h h constn= = = = + = = -B2: (Bin i)( )111( 1) 1 Smiim mimm mh k Ak A h mk Ah mh++ + | |= + = s |\ .Cng theo v cc BT (i = 1, n ) ta c: ( )0 0S S S const s =-B3:Xt du = 1 21 2......nnk A k A k A ha a a q+ = + = = + =+ + + =. VD4: Cho a,b,c >0 v a+b+c=6.Tm GTLN : 3 3 321 2 21 2 21 2 S a b b c c a = + + + + + + ++ Nhn xt:Ta c m=3;n=3;k=21 v q=6 62 21 2 2727.3a b c a b h = = = = + + = = Gii: p dng BT AM-GM ta c : ( ) ( ) ( )( ) ( ) ( )3 3 327.27 21 2 27.27 21 2 27.27 21 227.27 27.27 27.271 27 27 21 2 27 27 21 2 27 27 21 29 3 3 365 2 65 2 65 21. 9 (do6)9 3a b b c c aSa b b c c aa b b c c aa b c+ + + + ++= + ++ + + + + + + + + + ++| |s + + |\ .+ + + + + + ++= = + + = Du = a=b=c=2. Vy maxS=9a=b=c=2. Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 8 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - V.Dng 5: Cho cc s thc dng x (i = 1, n ) tha mn 1 21. ... hay( )nn iixx x q x q q const== = =[ .Tm GTNN ca: 1 21 2 1 1 2 1 1 2 1.... ... . ... . ...m m mnm m mx x xSAA A BB B CC C = + + + Trong : A], B], C] l cc tng i xng ca cc bin x ( ] = 1, m 1 ) PP chung : -B1:Ly k1, k2, , km-1 sao cho: 1 21 1 1 21 2 1 1 2 1....... ...nmmm mx x xA x A AAA A k k k = = == = = = - B2: (Bin i) p dng BT Cauchy (AM-GM) :1 1 1 2 11 2 1 1 2 11 2 11 2 1 2 21 2 1 1 2 11 2 1) .... .... ...) .... .... ........................................................................................mmmm mmmmmm mmA x A A mxAA A k k kkk kB x B B mxBB B k k kkk k + + + + + >+ + + + + >+1 1 21 2 1 1 2 11 2 1) .... .... ...mn m nmm mmx C mx C CCC C k k kkk k + + + + >Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 9 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Cng theo tng v ta c : 111 2 1 1 2 1 1 2 1. ... . ... . ...nnniiniim m mm m mn xxn qSkk k kk k kk k== > > =[ -B3: Xt du = 1 2...nx x x = = =. => KL : - VD5:Cho a,b,c >0 tha mn abc=8.Tm GTNN ca : 3 3 3( )( 2 ) ( )( 2 ) ( )( 2 )a b cSb c b c a c c a a b a b= + ++ + + + + + Nhn xt:Cho a=b=c , chn x , y : 32( )( 2 )a b c b cb c b c x y+ += =+ +122 318 6xa a ay x y= = = = Gii: p dng BT AM-GM ta c: 3 332 23 . .( )( 2 ) 12 18 ( )( 2 ) 12 18 2a b c b c a b c b c ab c b c b c b c+ + + ++ + > =+ + + + Tng t : 3 32 2; .( )( 2 ) 12 18 2 ( )( 2 ) 12 18 2b a c c a b c a b a b ca c c a a b a b+ + + ++ + > + + >+ + + +;Cng theo tng v cc BT trn ta c : ( )331 83 2 6 6a b c a b c a b c abcS S do abc+ + + + +++ > > > = = Du = a=b=c=2. Vy min S=1 a=b=c=2 Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 10 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - VI.Dng 6: Cho x,y,z > 0 v x +y +z k.Tm GTNN : m n pS ax by czx y z= + + + + + (a,b,c,m,n,p l cc hng s dng) PP chung : -B1:Chn x,y,z v cc hng s o1, b1, c1, k1 sao cho: 1 1 11 1 1 1; ;x y z km n pa x b x c xx y za a b b c c k + + == = = = = = -B2: (Bin i)( )1 1 1 1m n pS a x b y c z k x y zx y z| || | | |= + + + + + + + + |||\ . \ .\ . -B3: p dng BT AM-GM ( Cauchy) kt hp vi gi thit x +y +z k , chng minh 1 0S S >(0Sl hng s ) ri kt lun. Xt du ng thc xy ra khi no. VD6: Cho x,y,z > 0 v x +y +z S. Tm GTNN ca: 1 8 162 3 5 S x y zx y z= + + + + + Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 11 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Nhn xt:1 1 11 1 1 151 8 16; ;x y za x b x c xx y za a b b c c k + + == = = = = = Chn x=1;y=z=2 th 121 111;84 2 2;168 4 4 1x axy byz c kz= == = == = = = Gii: Ta c: ( )( )1 4 42 41 4 42 . 2.2 . 4.2 . 5 31 AM-GMmin 31 1; 2.S x y z x y zx y zx y z theox y zS x y z| || | | |= + + + + + + + + |||\ . \ .\ .> + + + = = = = = VII.Dng 7: Cho cc s thc o, b, c tha mn: o m; b n; p c p +1 v o2+b2+c2= k Trong m , n , p , k l cc hng s sao cho ( )22 210k p m nm n p= + + + < < < Tm GTNN ca biu thc : S = a + b + c . Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 12 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - PP chung : -B1: (t n ph) t x=a-m ; y=b-n ; z=c-p th x, y, z u -B2: (Bin i iu kin theo n ph) ( ) ( ) ( )2 2 22 2 22 2 2 2 2 22 2 2a b c k x m y n z p kx y z xm yn zp k m n p+ + = + + = + + + + + = -B3: (Ta s chng minh x+y+z ln hn hoc bng 1) Gi s x+y+z < 1 m x, y, z u nn x2+y2+z2< 1. Ta c : | |2 2 22 2 22 2 2 2 2 22 2 22 ( ) 2( )( ) 2( )1 2 2( ) 2( ) ( )1 2 ( ( 1) )x y z xm yn zpx y z mx y z n m y z p nzm n m p n x y zp k m n p Do k p m n+ + + + += + + + + + + + + < + + + + +< + = = + + +T ta c iu mu thun , suy ra x2+y2+z2 1. 1 a b c x y z m n p m n p ++= + ++ + + > ++ +. Vy min S=m+n+p+1 a=m ; b=n ; c=p+1 VD7: Cho cc s thc o, b, c tha mn: o 2; b S; 4 c S v o2+b2+c2= S8 Tm GTNN ca biu thc : S = a + b + c . Nhn xt: Ta s trnh by y ht pp chungGii: t x=a-2 ; y=b-3 ; z=c-4 th x, y, z u ( ) ( ) ( )2 2 22 2 22 2 238 2 3 4 384 6 8 9 (*)a b c x y zx y z x y z+ + = + + = + + + + + = Gi s x+y+z < 1 m x, y, z u nn x2+y2+z2< 1. Ta c : 2 2 22 2 24 6 84( ) 2( ) 2x y z x y zx y z x y z y z z+ + + + += + + + + + + + + Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 13 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | |1 4 2 2 ( ) 9 x y z < + ++ + + < Mu thun vi (*) , suy ra x2+y2+z2 1. 2 3 4 10 a b c x y z ++= + +++ + >. Vy min S=10 a=2 ; b=3 ; c=5 VIII.Dng 8: Cho a,b,c tha mn 2 2ma nb pab k + + s.Tm GTLN ca : S ax by abz = + + Trong m , n , p , k , x , y , z l cc hng s sao cho ( ) ( )2( ) 4 p z x m y nx my n = >Suy ra min f(t)=f(n) t=n v max f(t)=f(m)t=m. VD9: Cho t tha mn2 t >hay[2; ) t e + .CMR: 243 5tSt= >+ Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 15 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Gii: Xt hm s:2( )3tf tt=+ Xt t1, t2 bt k e |2; +) (do t 2) sao cho t1 > t2 > u . Ta c: (t1) (t2) = t1t2(t1 t2) +S(t12t22)(t1 +S)(t2 +S)> u Suy ra f(t) ng bin trn |2; +) suy ra (t) (2) =45 => pcm X.Dng 10: Cho cc s thc x, y > u ( 1, i n = ) tha mn 2 21 1n ni i ii ia x a y= =| |s |\ . CMR: 11ni ni iiiaax y==> PP chung : -B1: p dng BT Schwarz : 2111niniiniii iiaaxa x===| | |\ .> (*) -B2: p dng BT Bunhiacopxki: 2 2 22 21 1 1 1 1. .n n n n ni i i i i i i i ii i i i ia x a ax a a x y a= = = = =| | | | | || | | |= s s ||| ||\ . \ . \ .\ . \ . 11 1 1ni n n ni ii i ii i iiaaa x y ax y== = = s > (Th vo (*)) Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 16 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - B3: Ta c pcm . Du bng xy ra khi v ch khi 1 2...nx x x = = = VD10: Cho a,b,c,d > 0 tha mn 2 2 2 22 3 6 a b c d + + sCMR: 1 2 3 6a b c a+ + > (Bn c t gii) XI.Dng 11: Cho ia k > v 1niia m==(k l hng s sao cho mkn Gii bt phng trnh (**) ta c A > hoc A s

hoc A s s . -B2: Kt lun . Ch ra du bng xy ra khi no .VD12: Cho s thc t bt k. Tm GTNN , GTLN ca : 2221t tSt t +=+ + Gii: Ta c : ( )22221 (S 1) S 2 0 (*)1t tS S t tt t += + + + =+ + V tn ti gi tr ca S phng trnh (*) c nghim+) S =1 th phng trnh (*) c nghim +)S 1 =th phng trnh (*) l phng trnh bc hai nn Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 18 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ( ) ( )( )21 4 1 2 07 2 7 7 2 73 3S S SSA = + > + s s KL: 7 2 7 1 3 2 2SMin S tS += = Vi 7 2 7 3S= 7 2 7 1 3 2 2SMax S tS+ += =Vi 7 2 7 3S+=XIII.Dng 13: Cho hai s thc x,y tha mn 2 2ax by cxy k + + =Tm GTLN,GTNN (nu c) ca 2 2' 'y ' S ax b cxy = + +(Trong a,b,c,a,b,c,k l cc hng s) PP chung : (p dng dng 12) -B1: (Bin i) Ta c :2 22 2' 'y ' S ax b cxyAk ax by cxy+ += =+ + -B2: (nh gi) Xt hai trng hp : +) '0 .ay S ka= = +) 0 y =,chia c t v mu ca A cho y2 ri t t =x 22' ' ' S at ct bAk at ct b+ += =+ + -B3: Gii tip nh dng 12. Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 19 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - VD13: Cho hai s thc x , y tha mn 2 21 x y xy + + =Tm GTNN v GTLN ca 2 22 y S x xy = + Gii: Ta c: 2 22 22 y S= x xyx xy y ++ + +) 0 1 y S = = +) 0 y =,chia c t v mu ca A cho y2 ri t t =x 2221t tSt t +=+ + Ta c : ( )22221 (S 1) S 2 0 (*)1t tS S t tt t += + + + =+ + V tn ti gi tr ca S phng trnh (*) c nghim+) S =1 th phng trnh (*) c nghim +)S 1 =th phng trnh (*) l phng trnh bc hai nn ( ) ( )( )21 4 1 2 07 2 7 7 2 73 3S S SSA = + > + s s KL: 21.2 2 7 2 7 2( 1)37 6 3Sx ySMin SSyS S+ = = = + Vi 7 2 7 3S= 21.2 2 7 2 7 2( 1)37 6 3Sx ySMax SSyS S+= + = = +Vi 7 2 7 3S+= Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 20 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - XIV.Dng 14: Cho cc s thc a , b v cc hng s dng k , m tha mn 2 2 22 2 a b k ak bm + + = + .Tm GTNN , GTLN (nu c) ca: . . S a b = + ( , const = ) PP chung : -B1: (Bin i iu kin) ( ) ( )2 22 2 2 22 2 a b k ak bm a k b m m + + = + + = -B2: (Bin i) p dng BT Bunhiacopxki : ( ) ( ) ( ) ( ) ( )( ) ( )22 22 222 2 22 2 2 2a k b m a k b mS k m mk m m S k m m ( + + > + ( s + + + s s + + + -B3: Kt lun . Ch ra du bng xy ra khi no. VD14: Cho cc s a,b tha mn 2 29 6 4 a b a b + += + . Tm GTNN , GTLN ca: 3a 4 S b = +Gii: Ta c:( ) ( )2 22 29 6 4 3 2 4 a b a b a b + + = + + =p dng BT Bunhiacopxki : ( ) ( ) ( ) ( ) ( )( )22 22 223 2 3 4 3 3 4 217 10010 17 10 177 27a b a bSSS ( + + > + ( s + s s +s s Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 21 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Vy :min S=7 93 253 423 4 7 .5a b aa b b = = + = = max S=7 213 253 4183 4 27 .5a b aa b b = = + = = Hi Nhng Ngi Thch Ton L Ho - FACEBOOK ---------------------------------------------------------------------------------------------------------------- 22 | P a g e- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Ngy hon thnh : 8/6/2015 lc 22gi 20 pht . > -----o0o-----: The End :-----o0o-----