BDT_cuctri_LTĐH

8/9/2019 BDT_cuctri_LTH

1/12

DON XUN HUY-THPT n Thi-Hng Yn

BT NG THC V CC TR

chng minh cc BT ta c th s dng mt s bt ng thc hoc dng phng php nh gi

S dng mt s BT c bn:Cc BT c bn y l BT C-Si: Vi n s khng m bt k: 1 2; ;... ( 2)na a a n ta lun c:

1 21 2

...... ( )n n n

a a aa a a I

n

+ + + ; du bng xy ra khi v ch khi: 1 2 ... na a a= = = .

BT Bunhiacpxki: Vi hai b s thc bt k 1 2 1 2( ; ;... ),( ; ;... )n na a a b b b ta lun c:2 2 2 2 2 2 2

1 1 2 2 1 2 1 2... ) ( ... )( ... )( )n n n na b a b a b a a a b b b II + + + + + + + + + ; du bng xy ra khi v ch

Khi:

1 2

1 2 ...n

n

aa a

b b b= = = . BT:2 2 2

( )a b c ab bc ca III + + + + ; du bng xy ra khi a b c= =

T:2

1 2 1 2

1 1 1... ( )

...n n

nIV

a a a a a a+ + +

+ + +; trong 1 2, ,... na a a l cc s dng; du bng

y ra khi v ch khi cc s ny bng nhau.

Bi 1: Cho 0a b> > . Chng minh:

2 2

1 4 1/ 3; / 3; / 2 2.

( ) ( )( 1) ( )

a a b a c a

b a b a b b b a b

+ + +

+ Gii: a/ Theo BT (I) ta c: 3

1 1( ) 3 .( ). 3

( ) ( )b a b b a b

b a b b a b+ + =

(pcm).

Du bng xy ra khi 1; 2.b a= =

Bi 2: Cho a > 1; b > 1. Chng minh: 1 1 .a b b a ab +

Gii: Theo BT (I) ta c:( 1) 1

1 ( 1).1 .2 2

b aba b a b a

+ = = ; tng t ta cng c:

12

ab a . Cng cc v ca cc BT ny li ta s c pcm. Du bng xy ra khi a = b = 2.

Bi 2: a,b,c l ba s khng m c tng bng 1. Chng minh: 8/27ab bc ca abc+ + .

Gii: Theo BT (I) ta c: 3(1 ) (1 ) (1 ) 2

(1 )(1 )(1 )3 3

a b ca b c

+ + =

1 8/ 27a b c ab bc ca abc ab bc ca abc + + + = + + (pcm). Du bng xy ra kh

1


2/12

DON XUN HUY-THPT n Thi-Hng Yn= b = c =1/3.

Bi 3: Cho ba s khng m a,b,c. Chng minh: 3 3 3 2 2 2a b c a bc b ca c ab+ + + + .

Gii: Theo BT (I) ta c: ( )4

3 3 3 3 3 3 264 6 6a b c a b c a bc+ + = ; tng t ta cng c:3 3 3 2 3 3 3 24 6 ;4 6b c a b ca c a b c ab+ + + + cng cc v ca cc BT ny li ri n gin

c BT cn chng minh. Du bng xy ra khi a = b = c.Bi 3: Cho ba s dng x,y,z. Chng minh: 6 2 3( ) / 432 x y z xy z + + .Bi 4: Tm GTNN ca biu thc 9 3 6( ) /P x y x y= + trong x,y l cc s dng.

Gii: Theo BT (I) ta c:3 6 9 9 9

93 6 3 6 6

( ) 9 33. 6. 9.

3 6 3 6 3 6 2

x y x y x y x y P

x y

+ + = + = =

Vy GTNN ca P bng 9 63 / 2 khi y = 2x.Bi 5: Ba s thc a,b,c tha mn h thc: 6 6 6 3a b c+ + = . Hy tm GTLN ca biu thc

2 2 2S a b c= + +Gii: Theo BT (I) ta c: 6 2 6 2 6 21 1 3 ; 1 1 3 ; 1 1 3 9 3 3a a b b c c S S+ + + + + +

Vy GTLN ca S bng 3 khi a = b = c = 1.Bi 6: x,y l cc s thc tha mn cc iu kin: 0 3;0 4x y . Tm GTLN ca biu thc

(3 )(4 )(2 3 )A x y x y= + .

Gii: Theo BT (I) ta c: 3(6 2 ) (12 3 ) (2 3 )

2(3 ).3(4 ).(2 3 )3

x y x y x y x y

+ + + + =

3

6 6 36A A . Vy GTLN ca A bng 36 khi x = 0 v y = 2.Bi 7: x,y,z l cc s khng m c tng bng 1. Tm GTLN ca biu thc:( )( )( )P xyz x y y z z x= + + + .

Bi 8: a,b,c l cc s dng. Chng minh: *( , )m n m n m n

n n n

m m m

a b ca b c m n N

b c a

+ + +

+ + + +

Gii: Theo BT (I) ta c: ( ) ( ) ( )n

m n m nn n m n

m nm m

a an mb m n b m n a

b b

+ +

+

+ + = +

. Tng t

a cng c: ( ) ; ( )m n m n

n n n n

m mb cn mc m n b n ma m n cc a

+ +

+ + + + . Cng cc BT ny li ri n

in ta s c BT cn chng minh. Du bng xy ra khi a = b = c.

Ch : Nu 1m n= = th ta c BT:2 2 2

.a b c

a b cb c a

+ + + +

2


3/12


Bi 9: Cho 3 s thc dng a,b,c. Chng minh:3 3 3

.( ) ( ) ( ) 2

a b c a b c

b c a c a b a b c

+ ++ +

+ + +

Gii: Theo BT (I) ta c:3 3

33

3( ) 2 4 ( ) 2 4 2

a b c a a b c a a

b c a b c a

+ ++ + =

+ +. Tng t ta cng c

3 33 3;( ) 2 4 2 ( ) 2 4 2

b c a b b c a b c c

c a b a b c

+ ++ + + + + +

. Cng cc v ca cc BT ny li ri n

in ta s c BT cn chng minh. Du bng xy ra khi a = b = c.Bi 10: Cc s thc dng x,y,z tha mn iu kin: 6 x y z + + . Tm GTNN ca biu thc:

3 3 3 x y z S

y z x z y x= + +

+ + +.

Bi 11: Cho ba s thc dng a,b,c tha mn h thc: 6a b c+ + = . Tm GTNN ca biu thc:

3 3 31 1 1(1 )(1 )(1 )P a b c= + + + .

Bi 12: Cho x,y,z l ba s thc tho mn h thc: 0 x y z + + = . Chng minh:3 4 3 4 3 4 6 x y z S = + + + + +

Gii: Theo BT (I) ta c: 4 / 43 4 1 1 1 4 4 4 2.2 x x x x+ = + + + = . Tng t ta cng c:

3/ 4 / 4 / 4 / 4 / 4 ( ) / 43 4 2.2 ; 3 4 2.2 2(2 2 2 ) 2.3 2 6 y y z z x y z x y z S + ++ + + + = (pcm)

Du bng xy ra khi 0 x y z = = = .Bi 13: Cho hai s thc dng x,y c tng bng 1. Tm GTNN ca biu thc:

1 1

x yS

y x= +

.

Gii: D thy S dng. Theo BT (I) ta c:2 2

2 2 2x y

S x y xy xyy x

+ + + + +

2 2233. 3. 3( ) 2 2

x y xy xy x y S S

y x+ = + . Vy

2MinS =khi x = y = 1/2.

Bi 14: Cho ba s dng a,b,c tha mn iu kin: 3a b c+ + . Tm GTNN ca biu thc:a b c

Sb c a

= + + .

3


4/12

DON XUN HUY-THPT n Thi-Hng YnBi 15: Cho 3 s dng a,b,c tha mn h thc: 2 2 2 1.a b c+ + = Chng minh:

3ab bc ca

Sc a b

= + + .

Bi 16: Cho 3 s dng x,y,z c tng bng 1. Chng minh BT:3

2

xy yz zx

xy z yz x zx y+ + + + + .

Gii: Do ( ) ( )( ) xy z xy z x y z x z y z + = + + + = + + nn theo BT (I) ta c:1

.2

xy x y x y

xy z x z y z x z y z

= + + + + + +

. Tng t ta cng c:

1

2

yz y z

yz x x y x z

+ + + +

;1

2

xz x z

xz y x y y z

+ + + +

Cng cc BT trn ta s c BT cn chng minh. Du bng xy ra khi 1/ 3 x y z = = = .Bi 17: Cho hai s thc dng x,y tha mn iu kin: 6x y+ . Tm GTNN ca biu thc:

6 83 2 P x y

x y= + + + .

Gii: Theo BT (I) ta c:3 6 8 3 3 3 6 8 3

2. . 2. . .62 2 2 2 2 2 2

x y x y x yP

x y x y= + + + + + + +

6 4 9 19= + + = . Vy MinP = 19 khi x = 2 v y = 4.

Bi 18: Cho 3 s thc dng x,y,z tha mn iu kin: 2 1 xy xz + = . Tm GTNN ca biu th3 4 5 yz xz xyS

x y z = + + .

Gii: Theo BT (I) ta c: 2 3 2 4 6 yz xz yz xy xy xz

S z y x x y x z z y

= + + + + + + + =

2( ) 4( ) 4 8 4 x z x y xz xy+ + + + = . Vy MinS = 4 khi x = y = z = 1/3.

Bi 19: Cho hai s thc khng m x,y tha mn cc iu kin: 4;3 6 x y x y+ + .

m GTLN ca biu thc: 39. 4 P x y= + .

Gii: Theo BT (I) ta c:3 2 23.3 .1.1 .2 .3 3( 2) ( 3)

3 3 P x y x y= + + + +

2 3 3 9 2 3( ) (3 ) 6 2 3 4 6 6 2 3 4. 6. 6 2 3

2 6a x y b x y a b

= + + + + + + + + = + + +

9 4 3= + . ( Do 3 3 & 2/ 3 (2 3 3) / 2 & (9 2 3) / 6a b a b a b+ = + = = = ).4


5/12

DON XUN HUY-THPT n Thi-Hng YnVy 9 4 3MaxP= + khi 1& 3x y= = .

Bi 20: Cho 3 s dng a,b,c. Chng minh BT:1 1 1 1 1 1 1

2 2 2 4a b c a b c a b c a b c + + + + + + + + + +

.

Gii: Theo BT (IV) ng vi n =2 ta c: 1 1 1 1 12 ( ) ( ) 4a b c a b a c a b a c = + + + + + + + + 1 1 1 1 1 1 1 1 2 1 1

4 4 4 16a b a c a b c

+ + + = + + . Tng t ta cng c:

1

2a b c+ +1 1 2 1

16 a b c + +

; 1

2a b c+ +1 1 1 2

16 a b c + +

.Cng cc v ca cc BT ny li r

n gin ta s c BT cn chng minh. Du bng xy ra khi .a b c= =

Bi 21: Cho hai s dng a,b c tng bng 1. Chng minh cc BT sau:2 2 2 2

1 1 2 3/ 6; / 14.a b

ab a b ab a b+ +

+ +

Gii: a/ Theo BT (IV) ng vi n =2 ta c:2 2 2 2

1 1 1 1 1

2 2ab a b ab ab a b+ = + +

+ +

2 2 2

2 42 4 6

( ) 2a b ab a b+ = + =

+ + +(pcm). Du bng xy ra khi 1/2.a b= =

1/2.a b= =Bi 22: Cho a,b,c l cc s thc dng tha mn iu kin: 3/2.a b c+ + Chng minh:1/ 1/ 1/ 15/ 2.a b c a b c+ + + + +

Bi 23: Ba s dng x,y,z c tch bng 1. Chng minh: 2 2 2 x y z x y z + + + + .

Gii: p dng BT (II) v (I) ng vi n = 3 ta c:2

2 2 2 ( ) ( ).3

x y z x y z x y z

+ ++ + = + +

3( ).3

x y z x y z xyz x y z

+ + + + = + + (pcm). Du bng xy ra khi 1 x y z = = = .

Ch : T BT trn ta suy ra BT:2 2 2

2 2 2

a b c a b c

b c a b c a+ + + + vi a,b,c l cc s dng.

Bi 24: Cho 0; 0a c b c> > > > . Chng minh: ( ) ( )c b c c a c ab + .Gii: p dng BT (II) cho hai b s ( ; ) & ( ; )c a c b c c ta c:

2 ( ) ( )) ( )( )c b c c a c c a c b c c ab + + + = t suy ra BT ccm. Du bng xy ra

5


6/12

DON XUN HUY-THPT n Thi-Hng Yn( )ab c a b= +

Bi 25: Cho 4 s dng x,y,a,b tha man cc iu kin: ;a x a b x y> + > + . Chng minh:2 2 2( ) x a x a

x y a b x y a b

+

+ + +.

Gii: p dng BT (II) cho hai b s ; & ( ; ) x a x x y a b x yx y a b x y

+ + + + ta

c:2 2

2( ) ( ) ( ) x a x

x y a b x y x a xx y a b x y

+ + + + + + +

t suy ra BT ccm. D

ng xy ra khi bx = ay.Bi 26: Bn s thc a,b,c,d tha mn h thc: 2 2 2 2 1a b c d + + + = ; x l s thc bt k. Chngminh:

2 2 2 2 2 2

( ) ( ) (2 1) x ax b x cx d x+ + + + + +Gii: p dng BT (II) ng vi n = 3 ta c: 2 2 2 2 2 2 2 2( ) ( 1 )( ); x ax b x x x a b+ + + + + +

2 2 2 2 2 2 2 2) ( 1 )( )x cx d x x x c d + + + + + + 2 2 2 2( ) ( ) x ax b x cx d + + + + + 2 2 2 2 2 2 2 2 22 1)( ) (2 1)x x a b x c d x+ + + + + + = + (pcm). Du bng xy ra khi b=d=1&x=a=c

Bi 27: Cho 5 s dng x,y,z,p,q bt k. Chng minh:3 x y z

py qz pz qx px qy p q+ +

+ + + + .

Gii: Theo BT (III) ta c: ( ) ( ) ( ) ( )( ) x py qz y pz qx z px qy p q xy yz zx+ + + + + = + + +

2)( ) /3p q x y z + + + (*). p dng BT (II) cho hai b s ; ; x y z py qz pz qx px qy

+ + +

( ); ( ); ( )) x py qz y pz qx z px qy+ + + ta c:

[ ] 2( ) ( ) ( ) ( ) x y z

x py qz y pz qx z px qy x y z py qz pz qx px qy

+ + + + + + + + ++ + +

Kt hp vi BT (*) ta s c BT ccm. Du bng xy ra khi; py qz pz qx px qy+ = + = + .

Bng cch gii tng t ta s chng minh c cc BT sau:

1/3

2

a b c

b c a c b a+ +

+ + +vi a,b,c l cc s dng bt k.

2/ 2a b c d

b c d c d a a b+ + +

+ + + +vi a,b,c,d l cc s dng bt k.

6


7/12


3/2 2 2

2

a b c a b c

b c a c b a

+ ++ +

+ + +vi a,b,c l cc s dng bt k.

4/2 2 2a b c

a b cb c a a c b b a c

+ + + ++ + +

vi a,b,c l di ba cnh ca mt tam gic.

5/ 3a b cb c a a c b b a c+ + + + + vi a,b,c l di ba cnh ca mt tam gic.

Bi 28: Cho cc s thc x,y,u,v tha mn iu kin: 2 2 2 2 1 x y u y+ = + = . Chng minh:( ) ( ) 2u x y v x y + +

Gii: Theo BT (II) : [ ]2 2 2 2 2 2 2( ) ( ) ( ) ( ) ( ) 2( )u x y v x y u v x y x y x y + + + + + = + =

suy ra BT cn chng minh. Du bng xy ra khi ( ) ( ).u x y v x y+ = Bi 29: Cho a,b,c l 3 s dng tha mn iu kin: 2 2 2 1.a b c+ + Chng minh:

3 3 3 12

a b cb c a c b a

+ + + + +

Gii: Theo BT (II) ta c: [ ]3 3 3

( ) ( ) ( )a b c

a b c b a c c b ab c a c b a

+ + + + + + + + + +

2 2 2 2 2 2 2) ( )a b c a b c ab bc ca+ + + + + + . T ta suy ra BT cn chng minh. Du by ra khi 3 /3a b c= = = .

Bi 30: Ba s x,y,z tha mn iu kin: ( 1) ( 1) ( 1) 4 /3.x x y y z z + + Chng minh:

1 4 x y z + + .Gii: T iu kin ta suy ra: 2 2 2( 1/ 2) ( 1/ 2) ( 1/ 2) 25/12 x y z + + . p dng BT (II) ta

c:

]2 2 2 21.( 1/ 2) 1.( 1/ 2) 1.( 1/ 2) 3 ( 1/ 2) ( 1/ 2) ( 1/ 2) 25/ 4 x y z x y z + + + +

3/ 2 5/ 2 5/ 2 3/ 2 5/ 2 1 4 x y z x y z x y z + + + + + + (pcm).Du bng xy ra khi 4 /3 x y z = = = .Bi 31: Hai s a,b tha mn iu kin: 2 2 16 8 6a b a b+ + = + . Chng minh:

/10 4 3 40; / 7 24a a b b b a + Gii: a/ T iu kin ta suy ra: 2 2( 4) ( 3) 9a b + = . p dng BT (II) ta c:

]2 2 2 2 24( 4) 3( 3) ( 4) ( 3) (4 3 ) 9.25 4 3 25 15a b a b a b + + + = +

15 4 3 25 15 10 4 3 40a b a b + + (pcm). Du bng xy ra khi a = 24/5,b = 24/oc a = 16/5, b = 6/5.

7


8/12


Bi 32: Ba s x,y,z tha mn iu kin: 2 2 2 4 2 0. x y z x z + + + Tm GTNN v GTLN caiu thc:

2 3 2 .S x y z = + Bi 33: Cho a,b,c l ba s khng m tha mn h thc: 3.a b c+ + = Tm GTNN ca biu thc

2 2 2 2 2 2

S a ab b c cb b a ac c= + + + + + + + +.

Gii: Theo BT (II) ta c:2 22 2

2 2 2 24 3 1). 1 ( )3 2 2 2 23

b b b ba ab b a a a b

+ + = + + + + + = + 2 2 3( ) / 2a ab b a b + + + . Tng t ta cng c: 2 2 3( ) / 2c cb b c b+ + + ;

2 2 3( ) / 2 3( ) 3c ca a c a S a b c+ + + + + = . Vy MinS = 3 khi 3 /3a b c= = =

I.S dng phng php nh gi:

Bi 34: Cho 3 s dng a,b,c. Chng minh cc BT sau:

3 3 3 3 3 3

2 2 2

1 1 1 1/ ;

1 1 1/ .

2

aa b abc c b abc a c abc abc

a b cb

a bc b ac c ab abc

+ + + + + + + +

+ ++ +

+ + +Gii:a/Ta c: 3 3 2 2( )( ) ( ) ( )a b abc a b a ab b abc a b ab abc ab a b c

+ + = + + + + + = + + 3 3

1 1

( ) ( )

c

a b abc ab a b c abc a b c =

+ + + + + +. Tng t ta cng c cc BT:

3 3 3 3

1 1;

( ) ( )

a b

c b abc abc a b c c a abc abc a b c

+ + + + + + + +. Cng cc v ca cc BT n

i gin c ta s c BT cn chng minh. Du bng xy ra khi .a b c= =

b/ Theo BT (I) ta c:

2

2

1 1

2 0 2 42

bc b c

a bc a bc a bc abc abca bc

+

+ > = + .

ng t ta cng c:2 2

1 1;

4 4

a c b a

b ac abc c ab abc

+ +

+ +. Cng cc v ca cc BT ny li ri n

in ta s c BT cn chng minh. Du bng xy ra khi .a b c= =

Bi 35: Cho 3 s dng x,y,z tha mn iu kin: 2 2 2 3. x y z + + Tm GTNN ca biu thc:

8


9/12

DON XUN HUY-THPT n Thi-Hng Yn1 1 1

.1 1 1

P xy zy zx

= + ++ + +

Bi 36: Cho 3 s dng a,b,c c tng bng 2. Chng minh: 1.2 2 2

ab cb acS

c a b= + +

Bi 37: Cho 3 s dng a,b,c tha mn iu kin: 1/ 1/ 1/ 3.a b c+ + = Tm GTLN ca biu th3 3 3 3 3 3

.ab cb ac

Sa b c b a c

= + ++ + +

Bi 38: Cho ba s dng x,y,z c tch bng 8. Tm GTNN ca biu thc:2 2 2

2 2 2log 1 log 1 log 1.S x y z = + + + + +Gii: Ta

2 2 2

2 2 22 2 2

(log 1) (log 1) (log 1) 1( log 1 log 1 log

2 2 2 2

x x xS x y z

+ + + + + = + + + +

2

1 63 log 3 2.

2 2xyz + = = Vy 3 2MinS = khi 2. x y z = = =

Bi 39: Cho 3 s thc x,y,z c tng bng 1. Tm GTNN ca biu thc: 4 4 4 .S x y z xyz = + +

Gii: Theo BT (II) ta c:2

4 4 4 2 2 2 2 21 1 1 1( ) ( )3 3 3 27

x y z x y z x y z + + + + + + =

. p d

T (I) ta c:4 4 4

4 4 4

4

3 1 1 1/ 27 3.4

4 4 3 4.27 4 4 3

xyz x y z S x y z xyz

+ + = + + + + +

10.

4.27xyz xyz xyz = Vy 0MinS = khi 1/3. x y z = = =

Bi 40: Cho 3 s dng x,y,z bt k.Tm GTNN ca biuthc:2 2 2

2 2 2.

2 2 2

x y z S

x yz y yx z yx= + +

+ + +Bi 41: Cho 3 s dng x,y,z bt k. Chng minh:

4 6 4 6 4 6 4 4 4

2 2 2 1 1 1

.

x y z

S y z z x x y x y z = + + + ++ + +

II.Chng minh BT hoctm cc tr bng phng php i bin:

Bi 42: Cho cc s thc dng a,b,c tha mn h thc: .ab bc ca abc+ + = Chng minh BT:

9


10/12

DON XUN HUY-THPT n Thi-Hng Yn2 2 2 2 2 22 2 2

3b a c b a c

Sab cb ac

+ + += + + .

Gii: t x = 1/a, y = 1/b, z = 1/c th iu kin tr thnh: 1 x y z + + = v BT tr thnh:2 2 2 2 2 22 2 2 3S x y y z z x= + + + + + . Theo BT (II) ta c:

2 2 2( 2 ) /3 ( 2 ) /3 ( 2 ) /3 3( ) / 3 3S x y y z z x x y z + + + + + = + + = (pcm).Du bng xy ra khi 1/ 3 x y z = = = hay 3.a b c= = =Bi 43: Cho 3 s thc dng x,y,z c tch bng 1. Chng minh BT:

3 3 3

1 1 1 3.

( ) ( ) ( ) 2S

x y z y x z z y x= + +

+ + +Gii: t x = 1/a, y = 1/b, z = 1/c th iu kin tr thnh: 1abc = v BT tr thnh:

2 2 2 3

2

a b cS

b c a c b a

= + + + + +

.p dng BT (II)&(I) ta c ngay:2( )

2( ) 2

a b c a b cS

a b c

+ + + + =

+ +Du bng xy ra khi 1a b c= = = hay 1. x y z = = =Bi 44: Cho 3 s dng x,y,z tha mn iu kin: 1/ 1/ 1/ 1. x y z + + = Chng minh BT:

x yz y xz z yx xyz x y z + + + + + + + + .Gii: t x = 1/a, y = 1/b, z = 1/c th iu kin tr thnh: 1a b c+ + = v BT tr thnh:

1a bc b ac c ab ab bc ca+ + + + + + + + . Ta c:2 2( ) 2 ( )a bc a a b c bc a a bc bc a bc a bc+ = + + + + + = + = + . Tng t ta c

: ;b ac b ac c ab c ab+ + + + . Cng cc BT ny li ta s c BT ccm.Du bng xy ra khi 1/ 3a b c= = = hay 3. x y z = = =Bi 45: Cho hai s thc x,y khc 0 v tha mn iu kin: 2 2 2 22x y x y y x+ = + . Tm GTNN vGTLN ca biu thc: 2 / 1/ .S x y= +Gii: t 1/ & 1/u x v y= = th iu kin tr thnh:

2 2 2 22 ( 1/ 2) ( 1) 5/ 4 v u v u v+ = + + = . Theo BT (II) ta c:

[ ]22 2 2 2 22) 2( 1/ 2) 1 (2 1 ) ( 1/ 2) ( 1) 25/ 4 5/ 2 2S u v u v S = + + +

0,5 4,5S . Vy MinS = - 0,5 khi x = - 2; y = 2. MaxS = 4,5 khi x = y = 2/3.Bi 46: Hai s thc x,y tha mn cc iu kin: 20 & 12. y x x y + = + Tm GTNN v GTLN

ca biu thc: 2 17. A xy x y= + + +

Gii: T iu kin ta suy ra: 2 12 0 4 3 y x x x= + ;

ng thi 3 2( ) 3 9 7A f x x x x= = +

10


11/12

DON XUN HUY-THPT n Thi-Hng YnT BBT ca hm s ta suy ra:

( ) ( 3) (3) 20MaxA Maxf x f f = = = =[ ]4;3

( ) (1) 12MinA Minf x f = = =

[ ]4;3

Bi 47: Cho hai s dng x,y tha mn iu kin: 2 2 1x y+ = . Tm GTNN ca biu thc:( 1)(1 1/ ) ( 1)(1 1/ )S x y y x= + + + + +

Bi 48: Cho cc s thc x,y tha mn iu kin: 2 2 1x y+ = . Tm GTNN v GTLN

ca biu thc:2

2

4 2 1

2 2 3

x xyT

xy y

+ =

+

Gii: T iu kin ta suy ra:2 2

2 2

3 23 2

x xy yT

x xy y

+ =+ +

. Nu 20 1 1. y x T = = = Nu 0y

22

2

3 2 1/ (3 3) 2( 1) 1 0(*)

3 2 1

t t x y T T t T t T

t t

+ = = + + + =

+ +. (*) khng c nghim khi

Vi 1,(*)T c ' ( 1)( 2 4) 0T T = khi 2 1T < . Kt hp vi trn ta c:MinT=-2 khi 10 /10; 3 10 /10x y= =m . MaxT=1 khi 1x = v y = 0.Bi 49: Cho hai s dng x,y tha mn iu kin: 5/ 4x y+ = . Tm GTNN ca biu thc:

4 / 1/ 4 .S x y= +Bi 50: Cho hai s khng m x,y c tng bng 1. Tm GTNN v GTLN ca biu thc:

2008 20081 1S x y= + + + .

Gii: Ta c:

2007 20072008 2008

2008 2008

1004 1004(1 )( ) 1 1 (1 ) . '( )

1 1 (1 )

x xS f x x x f x

x x

= = + + + =

+ + 2007 2008 2007 2008 4014 2008'( ) 0 1 (1 ) (1 ) 1 1 (1 )f x x x x x x x = + = + + =

4014 2008 4014 4014 2008 2008 2006 2006

1 ) (1 ) (1 ) (1 ) (1 ) 0 x x x x x x x x + + = 2008 2008

1 2(2 1) ( ) (1 ) (2 1) ( ) 0 2 1 0 1/ 2x P x x x x P x x x + = = = .V x v 1 x khng ng thi bng 0 nn 1 2( ) 0; ( ) 0 P x P x> > )

Do 2008 2008(0) (1) 1 2; (1/ 2) 2 1 1/ 2 1 2; 2 1 1/ 2 f f f MaxS MinS = = + = + = + = +

x -4 -3 1 f(x) + 0 - 0 +f(x)

20 2

13 -12

11


12/12

DON XUN HUY-THPT n Thi-Hng YnTON V NH THC NIU-TN

Xc nh s hng trong khai trin ca nh thc Niu-tn:

Bi 1: Tm s hng khng cha x trong khai trin: 3 42 3 10 2 3 17/(2 3/ ) ; /(1/ )a x x b x x +

Gii: a/ Ta c:10 10

2 3 10 2 10 3 10 20 5

10 10

0 0

(2 3/ ) (2 ) (3/ ) 2 3k k k k k k k

k k

x x C x x C x

= =

= = .Do 20 5 0 4k k = = nn s hng khng cha x trong khai trin l 4 6 410 2 3 1.088.640.C =

b/ Ta c:17 17

3 42 3 17 2 / 3 17 3 / 4 (136 17 ) /12

17 17

0 0

(1/ ) ( ) ( )k k k k k

k k

x x C x x C x

= =

+ = = .Do 136 17 0 8k k = = nn s hng khng cha x trong khai trin l 817 24.310C = .

Bi 2: Bit h s ca s hng th 3 trong khai trin nh thc2 3

( / )n

x x x x+ bng 36. Tm s hh 7.Gii: T GT 2 ( 1) / 2 36 8nC n n n = + = = . Vy s hng th 7 trong khai trin bng

6 5 / 2 2 2 / 3 6

8 ( ) ( ) 28 .C x x x =

Bi 3: Tm h s ca 4x trong khai trin ca:2 10(1 2 3 )x x+ + .

Gii: Ta c:10 10

102 10 2 2 10 2 10

10 10

0 0 0

1 2 3 ) (1 2 ) 3 (1 2 ) (3 ) (2 ) (3 )k

k k k k l l k

k

k k l

x x x x C x x C C x x

= = =

+ + = + + = + = 10 20 2

10

10

3 2 .20 2 4 2 16.k l k l l k kl k

C C x l k l k +

+ = = C 3 b s (k;l) tha mn h thc n: (8;0), (9;2) v (10;4). Vy h s ca 4x bng:

8 2 9 2 2 4 4

10 10 9 10.3 . .2 .3 .2 8.085.C C C C + + =Bi 4: Tm h s ca 10x trong khai trin ca:

2 3 5(1 ) x x x+ + + .

Gii: Ta c:2 3 5 5 2 5 2

5 5

0 ; 5

(1 ) (1 ) (1 ) . 2 10k l k l

k l

x x x x x C C x k l +

+ + + = + + = + = c 3 cp (k;l) tmn l: (0;5), (2;4) v (4;3). Vy h s ca 10x trong khai trin bng:

0 5 2 4 4 3

5 5 5 5 5 5. . . 1C C C C C C + + =

12

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