Tuyển Chọn 50 Bài Toán Điển Hình Min-max

7/23/2019 Tuyn Chn 50 Bi Ton in Hnh Min-max

1/28

TUY N CHN 50 B I TO N I N H NH

MIN - MAX

CM NANG CHO MA THI

NGUY N H U BI N

https://www.facebook.com/ng.huubienEmail: [email protected]

(N THI THPT QUC GIA)


2/28

TUYN TP 50 BI TON IN HNH V MIN - MAX TRONG CC THI TH THPT QUC GIA 2015

Trang 1NGUYN HU BIN -https://www.facebook.com/groups/nguyenhuubien

Bi 1:Cho ba sdng x, y, z tha mn: 1x y z+ + =

Tm gi trnhnht ca:x y y z z x

Pxy z yz x zx y

+ + += + +

+ + +

Hng dn

Ta c 1 1+ + = + =

x y z x y z , ta c:1 11 (1 )(1 )

+ = =

+ +

x y z z

xy z xy x y x y

1 1

1 (1 )(1 )

+ = =

+ +

y z x x

yz x yz y z y z

1 1

1 (1 )(1 )

+ = =

+ +

z x y y

zx y zx x z x z

Khi + + +

= + ++ + +

x y y z z xP

xy z yz x zx y= 1

(1 )(1 )

z

x y+ 1

(1 )(1 )

x

y z+ 1

(1 )(1 )

y

x z

3 1 1 13 . . 3(1 )(1 ) (1 )(1 ) (1 )(1 )

=

z x yx y y z x z

.

Vy 3=MinP t c khi1

3= = =x y z

Bi 2: Cho x, y, z l ba sthc ty tha mn x + y + z = 3.

Chng minh rng vi 1a ta lun c :1 1 1

.x y z x y z

x y z

a a a a a a+ + + +

Hng dn

* Vi a = 1 ta thy BT ng .

* Ta xt khi a > 1.

Hm sy =1 1

t

ty

a a

= =

nghch bin vi t R , khi a > 1.

Khi ta c

Ta c :1 1

( )( ) 0,x y

x ya a

, .x y R Suy rax y y x

x y x y

a a a a+ + (1)

Chng minh tng t y z y zy z z ya a a a

+ + (2) z x z xz x x za a a a

+ + (3)

Cng vvi v(1) ,(2) v (3) ta c 2( )x y z x y z

x y z y z z x x y

a a a a a a

+ + ++ + + + (4)

Cng 2 vca (4) vi biu thcx y z

x y z

a a a+ + ta c

1 1 13( ) ( )( )

x y z x y z x y z

x y z x y z x y z x y zx y z

a a a a a a a a a

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


3/28



Suy ra1 1 1

.x y z x y z

x y z

a a a a a a+ + + + ( do x + y + z = 3 )

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

Bi 3:Cho , ,a b c l ba sthc dng tha mn iu kin 3.ab bc ca+ + =

Chng minh rng:2 2 2

1 1 1 1.

1 ( ) 1 ( ) 1 ( )a b c b c a c a b abc

+ +

+ + + + + +

Hng dn

p dng BT Cauchy cho 3 sdng ta c: 233 3 ( ) 1ab bc ca abc abc= + + .

Suy ra: 2 22

1 11 ( ) ( ) ( ) 3 (1).

1 ( ) 3a b c abc a b c a ab bc ca a

a b c a+ + + + = + + =

+ +

Tng tta c:2 2

1 1 1 1(2), (3).

1 ( ) 3 1 ( ) 3b c a b c a b c

+ + + +

Cng (1), (2) v (3) theo vvi vta c:

2 2 2

1 1 1 1 1 1 1 1

( )1 ( ) 1 ( ) 1 ( ) 3 3

ab bc ca

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

+ +

+ + + + = =+ + + + + + .

Du = xy ra khi v chkhi 1, 3 1, ( , , 0).abc ab bc ca a b c a b c= + + = = = = >

Bi 4: Cho x, y, z l cc s thc tha mn 0,0,221221 >>+


4/28



Do 2222 )1(4

4

4)1(

1

1

1

1

)1(

1

)1(

1

xxyzzy ++=

++

+

+

++

22 )1(8

1

)1(4

4

++

++

xxP

Do 221221 + 1. Tm gi trnhnht ca( ) ( )3 3 2 2

( 1)( 1)

x y x yP

x y

+ +=

Hng dn

t t = x + y ; t > 2. p dng BT 4xy (x + y)2ta c2

4

txy

3 2 (3 2)

1t t xy t

P xy t

= + . Do 3t - 2 > 0 v

2

4

txy nn ta c

23 2

2

2

(3 2)

42

14

t tt t

tP

t tt

=

+

Xt hm s2 2

2

4( ) ; '( ) ;

2 ( 2)

t t tf t f t

t t

= =

f(t) = 0 t = 0 v t = 4.


5/28



t 2 4 +f(t) - 0 +

f(t)

+ +

8

Do min P =(2; )min ( )f t

+

= f(4) = 8 t c khi 4 24 2

x y x

xy y

+ = =

= =

Bi 6 : Cho a,b,c l cc s dng tha mn a + b + c = 1.

Chng minh rng: 3a b b c c a

ab c bc a ca b

+ + ++ +

+ + +

Hng dn

* Bin i1 1

1 (1 )(1 )

a b c c

ab c ab b a a b

+ = =

+ +

* T 1 1 1(1 )(1 ) (1 )(1 ) (1 )(1 )

c b a

VTa b c a c b

= + +

Do a,b,c dng v a+b+c=1 nn a,b,c thuc khong (0;1) => 1-a,1-b,1-c dng* p dng bt ng thc Csi cho ba sdng ta c

31 1 1

3. . .(1 )(1 ) (1 )(1 ) (1 )(1 )

c b aVT

a b c a c b

=3 (pcm)

ng thc xy ra khi v chkhi1

3a b c= = =

Bi 7:Cho cc sthc dng x, y, z tha mn: 1yz zx xy

x y z+ + = .

Tm gi trln nht ca biu thc:1 1 1

1 1 1A

x y z= + +

.

Hng dn

t , ,yz zx xy

a b cx y z

= = = . Ta c a, b, c > 0 v 2 2 2 1a b c+ + = . Ta c:

1 1 1

31 1 1 1 1 1

bc ca ab

A bc ca ab bc ca ab= + + = + + + . Dc:

( )( )

2

2 2 2

2 2 2 2 2 2 2 2 2 2

1 141 2 2

12

b cb cbc b c

bc b c b a c a b a c a

+

+ = +

+ + + + + +


6/28



Tng tc:2 2

2 2 2 2

1

1 2

ca c a

ca c b a b

+

+ + v

2 2

2 2 2 2

1

1 2

ab a b

ab a c b c

+

+ +

t: A3 9

32 2

+ = . Du bng xy ra khi x = y = z =1/3

Bi 8: Cho , ,a b c l cc sthc dng v 3a b c+ + = .

Tm gi trln nht ca biu thc( )( )( )

32

3 1 1 1

abcP

ab bc ca a b c= +

+ + + + + +

Hng dnp dng Bt ng thc: 2( ) 3( )x y z xy yz zx+ + + + , , ,x y z ta c:

2( ) 3 ( ) 9 0ab bc ca abc a b c abc+ + + + = > 3ab bc ca abc + +

Ta c: 33(1 )(1 )(1 ) (1 ) , , , 0a b c abc a b c+ + + + > . Tht vy:

( ) ( ) ( ) 2 33 331 1 1 1 ( ) ( ) 1 3 3 ( ) (1 )a b c a b c ab bc ca abc abc abc abc abc+ + + = + + + + + + + + + + = +

Khi :3

32

3(1 ) 1abc

P Qabc abc

+ =+ +

(1).

t 6 abc t = ; v a, b, c > 0 nn3

0 13

a b cabc

+ + < =

Xt hm s ( ]2

3 2

2, 0;1

3(1 ) 1

tQ t

t t= +

+ +

( )( )

( ) ( )( ]

5

2 23 2

2 1 1( ) 0, 0;1

1 1

t t tQ t t

t t

=

+ +

.

Do hm sng bin trn ( ]0;1 ( ) ( )1

16

Q Q t Q = = (2). T(1) v (2):1

6P .

Vy maxP = 16

, t c khi v v chi khi : 1a b c= = = .

Bi 9: Cho , ,a b c l cc sdng v 3a b c+ + = .


bc ca ab

a bc b ca c abP + +

+ + +=

Hng dn

V a + b + c = 3 ta c3 ( ) ( )( )

bc bc bc

a bc a a b c bc a b a c= =

+ + + + + +

1 1

2

bc

a b a c

+

+ +

V theo BT C-Si:1 1 2

( )( )a b a c a b a c+ + + + + , du ng thc xy ra b = c

Tng t1 1

23

ca ca

b a b cb ca

+

+ ++ v

1 1

23

ab ab

c a c bc ab

+

+ ++

Suy ra P3

2( ) 2( ) 2( ) 2 2

bc ca ab bc ab ca a b c

a b c a b c

+ + + + + + + = =

+ + +,

ng thc xy ra khi v chkhi a = b = c = 1. Vy max P =3

2khi a = b = c = 1.


7/28



Bi 10:Cho a, b, c l cc sdng v 3a b c+ + = . Tm gi trln nht ca biu thc

3 3 3

bc ca abP

a bc b ca c ab= + +

+ + +.

Hng dn

V a + b + c = 3 ta c 3 ( ) ( )( )

bc bc bc


+ + + + + +

1 1

2

bc

a b a c

+

+ +


( )( )a b a c a b a c+

+ + + +, du ng thc xy ra b = c

Tng t1 1

23

ca ca

b a b cb ca

+

+ ++ v

1 1

23

ab ab

c a c bc ab

+

+ ++

Suy ra P3

2( ) 2( ) 2( ) 2 2


a b c a b c

+ + + + + + + = =

+ + +,


2

khi a = b = c = 1.

Bi 11:Cho ba sthc khng m a, b, c tha mn: a2009+ b2009+ c2009= 3.Tm gi trln nht ca biu thc: P = a4+ b4+ c4.

Hng dnp dng bt ng thc C-si cho 2005 s1 v 4 sa2009ta c:

20092009 2009 2009 2009 2009 2009 2009 2009 4

2005

1 1 ... 1 2009. . . . 2009. (1)+ + + + + + + =

a a a a a a a a a

Tng t: 20092009 2009 2009 2009 2009 2009 2009 2009 42005

1 1 ... 1 2009. . . . 2009. (2)+ + + + + + + =

b b b b b b b b b

20092009 2009 2009 2009 2009 2009 2009 2009 4

20051 1 ... 1 2009. . . . 2009. (3)+ + + + + + + =

c c c c c c c c c

T(1), (2), (3) ta c: 2009 2009 2009 4 4 46015 4( ) 2009( )+ + + + +a b c a b c

4 4 46027 2009( ) + +a b c . T suy ra 4 4 4 3= + + P a b c Mt khc ti a = b = c = 1 th P = 3 nn gi trln nht ca P = 3.

Bi 12:Cho x, y, z 0 thomn x + y + z > 0.

Tm gi trnhnht ca biu thc( )

3 3 3

3

16x y zP

x y z

+ +=

+ +

Hng dn

Trc ht ta c:( )

3

3 3

4

x yx y

++ (bin i tng ng) ( ) ( )

2... 0x y x y +

t x + y + z = a. Khi ( ) ( )

( )3 33 3

3 33 3

64 644 1 64

x y z a z zP t t

a a

+ + + = = +

(vi t =z

a, 0 1t )


8/28



Xt hm sf(t) = (1 t)3+ 64t3vi t [ ]0;1 . C

( ) [ ]22 1'( ) 3 64 1 , '( ) 0 0;1

9f t t t f t t = = =

Lp bng bin thin

( )[ ]0;1

64inf

81tM t

= GTNN ca P l16

81

t c khi x = y = 4z > 0

Bi 13: Cho ba sdng x,y,z tha x + y + z = 4 v xyz = 2.Tm GTNN ca biu thc: P = x4+ y4+ z4

Hng dn

( )

( ) ( ) ( ) ( )

( ) ( )

22 2 2 2 2 2 2 2 2

22 2

2

2( )

= 2 2 2

= 16 2 2 16

P x y z x y y z z x

x y z xy yz zx xy yz zx xyz x y z

xy yz zx xy yz zx

= + + + +

+ + + + + + + +

+ + + +

i

i t t = xy + yz + zx = x(y + z) + yz+ Tgt

24 ,y z x yz

x + = = ( ) 2

2 24 4t x x x x

x x = + = + +

+ Ta c: ( )22 3 28( ) 4 4 8 16 8 0y z yz x x x x

x+ +

( )( )22 6 4 0x x x + (*)

Gii BT (*) giao vi iu kin 0 < x < 4 ta c: 3 5 2x

+ Kho st hm st theo bin x vi 3 5 2x ta tm c:5 5 1

52

t

i ( )2 2 2

16 2 2( 16) 2 64 288P t t t t = = +

Kho st hm s: f(t) = 2t2 64t + 288 vi5 5 1

52

t

ta c:

5 5 1M inf( ) 383 165 5 khi , ( ) 18 khi 5

2t t Maxf t t

= = = =

Suy ra: min 383 165 5P = t c chng hn1 5

3 5,2

x y z +

= = =

max 18P = t c chng hn khi x = 2, y = z = 1

Bi 14: Cho cc sthc ;x y thay i.

Tm gi trnhnht ca biu thc: 2 2 2 22 1 2 1 2P x y x x y x y= + + + + + + + .

Hng dn2 2 2 22 1 2 1 2P x y x x y x y= + + + + + + +

Xt cc imM(x1; y),N(x+1; y). Ta c OM + ON MN

2 2 2 2 2( 1) ( 1) 4 4x y x y y + + + + +

22 1 2 ( )P y y f y + + =


9/28



TH1: y 2: 2( ) 2 1 2f y y y= + + 2

2'( ) 1

1

yf y

y=

+

22

0 3'( ) 0 2 1

33 1

yf y y y y

y

= = + =

=

Lp bng bin thin f(y) ( .2]3

min ( ) 2 33x f y f

= = +

TH2: y 2: 2( ) 2 1 2f y y y= + + 2 5 2 3> +

Vy 2 3 ;P x y + .

Do 2 3MinP = + khix = 0 ;y =3

3

Bi 15:Cho cc sthc dng a,b,c tha a + b + c =3. Tnh gc gi trnhnht ca biu

thc2 2 2

a bc b ca c abP

b ca c ab a bc

+ + += + +

+ + + Hng dn

Xt2 2 21 a bc b ca c ab

P3 3b 3ca 3c 3ab 3a 3bc

+ + += + +

+ + +

Ta c 3b 3ca b(a b c) 3ca b(a b c) ca 2ca+ = + + + = + + + + m 2 2a c 2ac+ nn 2 2 23b 3ca ab b bc ca a c+ + + + + + Chng minh tng tta c: 2 2 23c 3ab ac c bc ab a b+ + + + + +

2 2 23a 3bc a ab ac bc c b+ + + + + +

Khi 2 2 2

2 2 2

1 a bc b ca c abP 1 P 3

3 ab b bc ca a c

+ + + + + =

+ + + + +

Du = xy ra khi a = b = c = 1.Vy MinP 3= khi a = b = c = 1.Bi 16:Cho 3 sdng x, y, z tha mn xy + yz + zx = 3xyz.

Chng minh rng :3 3 2 2 3 3 2 2 3 3 2 2

3

4

xy yz zx

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

+ + + + + + + + +

Hng dn

Ta c : xy + yz + zx = 3xyz1 1 1

3 + + =x y z

Vi x >0; y > 0; z > 0 ta c x3+ y3xy(x + y) ; 1 1 1 1( )4 ++x y x y ;x2+ y22xy

3 3 2 2 2 2 2 2

1 1

4

xy xy xy

xy( x y )x y x z y z xy(x y) (x y )z (x y )z

+

++ + + + + + +

3 3 2 2 2 2

1 1 1 1 1

4 4 2

xy xy

(x y) (x y) zx y x z y z (x y )z

+ +

+ ++ + + +


10/28



1 1 1 1 1 1 1 1 1

4 4 2 16 8x y z x y z

+ + = + +

(1)

Chng minh tng t:

3 3 2 2

1 1 1 1

16 8

yz

y z xy z y x z x

+ +

+ + + (2)

3 3 2 2 1 1 1 116 8zx z x yz x z y x y

+ + + + + (3)

Cng (1) ; (2); (3) theo vta c pcm

ng thc xy ra khi x = y = z = 1

Bi 17:Cho cc sthc dng , ,x y ztha mn 2 2 25( ) 9( 2 )x y z xy yz zx + + = + + .

Tm gi trln nht ca biu thc2 2 3

1

( )

xP

y z x y z =

+ + +

.

Hng dnTheo githit ta c+ + = + + + + = + + + + +

2 2 2 25( ) 9( 2 ) 5( ) 9( 2 ) 10( )x y z xy yz zx x y z xy yz zx xy yz zx

+ + = + + + + +2 25( ) 19 ( ) 28 19 ( ) 7( )x y z x y z yz x y z y z

+ + +

+ + +

195 1 7 2 2( )

x x xx y z

y z y z y z

Mt khc ta c + + + +2 2 2 2 2 21

( ) 2( ) ( )2

y z y z y z y z

V vy( )

+ =

+ ++ + ++3 32

2( ) 1 4 1

1 27( )2( )( )2

y zP

y z y zy z y z y z

t +

= + > = + 2

3 3

4 1 (6 1) (2 1)0 16 16

27 27

t tt y z P

t t t

Vy =min 16P ; du bng t ti

= +

= =

= = + =

12( )3

11

126

x y z xy z

y zy z

Bi 18:Cho cc sthc dng ,x y tha mn1

3 ln 9 3 3 .3

x yxy x y

xy

+ ++ =

Tm gi trln nht ca biu thc:2 2

3 3 1 1 1

( 1) ( 1)

x yM

y x x y x y x y= + +

+ + +

Hng dnTgithit ta suy ra ln( 1) 3( 1) ln(3 ) 3.3x y x y xy xy+ + + + + = + .


11/28



Xt hm s ( ) ln 3g t t t = + trn (0; )+ , ta c1

'( ) 3 0g tt

= + > vi 0t > , suy ra ( )g t

ng bin trn (0; )+ , t ( 1) (3 ) 1 3g x y g xy x y xy+ + = + + = (*)

Theo (*) ta c 3 1 2xy x y xy = + . t 3 2 1 0 1.t xy t t t = 2 2 2

2

3 3 3 ( 1) 3 ( 1) 36 27 3.

( 1) ( 1) ( 1) 4

x y x y y x t t

y x x y xy xy x y t

+ + + ++ = =

+ + + + +

(2)

2 2 2 2

2 2 2 2 2 2

1 1 (3 1) 2 36 32 4

4

x y t t t t

x y x y t t

+ + = = = (3)

Theo C si1 1 1

22x y xy

+(4). T(2), (3), (4) ta c

2

5 1 1

4 2

tM

t

+ .

Xt hm s2

5 1( )

4

tf t

t

= trn [1;+ ) , ta c

2

4 3

5.4 (5 1)8 2 5'( ) 0 1

16 4

t t t t f t t

t t

= = < , suy ra ( )f t nghch bin trn [1;+ ) , bi vy

max[1; )

3max ( ) (1) 1 1.

2M f t f t x y

+

= = = = = =

Bi 19:Cho x, y, z l cc sthc dng v tha mn: ( ) 1z z x y x y = + + .

Chng minh rng :4 4 6

3 9

3

( ).( ).( ) 4

x y

x yz y zx z xy

+ + +.

Hng dnV ( ) 1z z x y x y = + + (z + 1)( x + y) = z

2- 1 v do z > 0 nn ta c: zyx =++ 1 .

Khi T =[ ]3

44

)1)(1().1).().(1).(( ++++++ yxxyxyyx

yx =[ ]42

44

)1)(1(.)( +++ yxyx

yx

p dng BT Csi cho cc sdng x, y ta c :

( )27

.427

41333

13

4

4

4

344 xxxxx

x =

+++=+ ;

( )27

.427

41333

13

4

4

4

344 yyyyy

y =

+++=+ ; ( ) xyyx 42 + .

Do [ ]42 )1)(1(.)( +++ yxyx 446

9

6

338 ..

3

4

3

..4.4 yx

yxxy = suy ra

9

6

4

3T ( * )

Du = ( * ) xy ra 7,3,31

133 ===

++=

== zyx

yxz

yx

.

Vy bt ng thc c chng minh.


12/28



Bi 20:Chox,yl hai sthc tha mn iu kin 24)( 3 ++ xyyx .

Tm gi trnhnht ca biu thc 2015)43()(2)(3 2222 +++= xyxyyxyxP .

Hng dnVi mi sthc x, y ta lun c 2(x y) 4xy+ , nn tiu kin suy ra

3 2 3 3 2( ) ( ) ( ) 4 2 ( ) ( ) 2 0 1+ + + + + + + + + x y x y x y xy x y x y x y

Ta bin i P nhsau 2 2 2 2 2 2 2 23 3

P (x y ) (x y ) 2(x y 2xy) xy(3xy 4) 20152 2

= + + + + + +

2 2 2 4 4 2 23 3(x y ) (x y ) 2(x y ) 20152 2

= + + + + + (3)

Do2 2 2

4 4 (x y )x y2

++ nn t(3) suy ra 2 2 2 2 2

9P (x y ) 2(x y ) 2015

4 + + +

t 2 2x y t+ = th1

t2

(do x y 1)+ .

Xt hm s 29f (t) t 2t 20154= + vi 1t 2

, c 9f '(t) t 2 02= > , vi 1t 2

nn hm s

f(t) ng bin trn1

;2

+

. Suy ra1

t ;2

1 32233min f (t) f

2 16 +

= =

.

Do GTNN ca P bng16

32233, t c khi v chkhi

2

1==yx

Bi 21:Cho a, b, c l di 3 cnh ca mt tam gic.

Chng minh rng:2

2.

3 3 2 3 3

a a a b c

a b a c a b c a c a b

+ + + + + > + > .

T(1),(2) v (3) ta c 2 2 2 2x y z x y zy z z x x y x y z

+ ++ + < =

+ + + + +(pcm).

+) t ; ; ( , , 0).2 2

a b c ax y z a x y z

+ += = = > Ta c: ; ;x y z y z x z x y+ > + > + > .

VT = 2 2 2 23 3 2 2 2 2 2 2 2

a c a b a x y z x y z

a b a c a b c y z z x x y y z z x x y

+ ++ + = + + = + +

+ + + + + + + + + +(1).

Li c:2z

( ) 2z( ) z

x y z z x y z x y x y z x y+ > + + < + >+ + + .

CM tng tta c: 2 2(2); (3).x x y yy z x y z z x x y z

< 2 2 2 2 8

12 3 27 8

b b

ab bc aca b c

+ + ++ + +

Suy ra( )

( )

2 2 8

1 2 2 8 2 8

2 2 8

1 2 2 8

ab bc ac b bP

ab bc ac ab bc ac ab bc ac

ab bc acP

ab bc ac ab bc ac

+ + + +

+ + + + + + + + +

+ + +

+ + + + + +

t t [ ]2 0;13ab bc ac t = + +


14/28



Xt hm s ( ) [ ]2 8

, 0;131 8

tf t t

t t= +

+ +

( )( ) ( )

( )2 22 8

' , ' 0 61 8

f t f t tt t

= = =+ +

( ) ( ) ( ) ( ) [ ]16 47 16

0 1; 6 ; 13 0;137 21 7

f f f f t t= = =

Do :16

7P . Khi

21; 2;

3a b c= = = th

16

7P = . Vy gi trln nht ca P l

16

7

Bi 24:Cho x l sthc thuc on5

[ 1, ]4

.

Tm gi trln nht, gi trnhnht ca5 4 1

5 4 2 1 6

x xP

x x

+=

+ + +

Hng dn

t 5 4 , 1a x b x= = + th 2 24 9,a b+ = vi , 0a b

Do t [0, ]2 vi a=3sin ,2b=3cos . Khi :

33sin cos 2sin cos2

2 6 3sin 3cos 6 2sin 2cos 4

a bP

a b

= = =+ + + + + +

Xt hm s2sin cos

( )2sin 2cos 4

x xf x

x x

=

+ +vi [0, ]

2x

Ta c /2

6 4sin 8cos( ) 0, [0, ]

(2sin 2cos 4) 2

x xf x x

x x

+ += >

+ +

Suy ra hm sf(x) lun lun ng bin trn [0, ]2

Do :[0, ] [0, ]

2 2

1 1min ( ) (0) ;max ( ) ( )

6 2 3x x

f x f f x f

= = = =

Vy1 5

min6 4

P khi x

= =

1 1

3Max P khi x= =

Bi 25: Cho 3 sthc dng , ,a b c thomn 1abc = .

Chng minh rng: 12 2 2a b c

b a c b a c+ +

+ + +.

Hng dn

Ta c12 2

a a a

a bab a a ba=

+ ++ +, do 1 2a a+ .

Tng t:12

b b

b bcc b

+ ++;

12

c c

c aca c

+ ++.

Cng cc vca cc BT trn ta c:


15/28



1 1 12 2 2

a b c a b c

a ba b cb c acb a c b a c+ + + +

+ + + + + ++ + +

=1

abc b cb

bc bca babc b cb b bc bac+ +

+ + + + + +

=1

11 1 1

b cb

bc b b cb b bc+ + =

+ + + + + +(iu phi chng minh).

Du bng xy ra khi v chkhi a = b = c = 1

Bi 26:Cho a, b, c l cc sthc dng thomn a b c 3+ + = .

Tm gi trln nht ca biu thc( )( )( )

32

3 1 1 1

abcP

ab bc ca a b c= +

+ + + + + +

Hng dn

p dng Bt ng thc ( ) ( )2

3 , , ,x y z xy yz zx x y z+ + + + ta c:

( ) ( )2

3 9abc 0ab bc ca abc a b c+ + + + = >

3ab bc ca abc + +

Ta c: ( )( )( ) ( )3

31 1 1 1 , , , 0.a b c abc a b c+ + + + > Tht vy:

( )( )( ) ( ) ( )1 1 1 1a b c a b c ab bc ca abc+ + + = + + + + + + +

( ) ( )323 331 3 3 abc 1abc abc abc+ + + = +

Khi ( )

( )3

3

2 1

13 1

abcP Q

abcabc + =

++

t 6 abc t = . V , , 0a b c > nn3

0 13

a b cabc + + < =

Xt hm s( )

( ]2

23

2, t 0;1

13 1

tQ

tt= +

++

( ) ( )( )

( ) ( )( ]

5

2 23 2

2 1 1' 0, t 0;1

1 1

t t tQ t

t t

=

+ +

Do hm sng bin trn ( ]0;1 nn ( ) ( ) ( )5

1 26

Q Q t Q= =

T(1) v (2) suy ra5

6P

Vy5

max6

P= , t c khi v chkhi: 1a b c= = = .

Bi 27: Cho 3 sthc , ,x y z khc 0 tha mn: x 5y z+ + = v . . 1x y z = .Tm gi trln

nht ca biu thc: 1 1 1Px y z

= + + .


16/28



Hng dn

( )1 1 1 1 1

5y z

P x xx y z x yz x

+= + + = + = +

Ta c: ( ) ( )2 2 4

4 5 0 3 2 2 4 3 2 2y z yz x x x xx

+ < +

Xt hm s: ( ) ( ) ( )2

1 15 5 2xf x x x f ' x

x x= + = +

Vi: 0 3 2 2 4 3 2 2x x x< +

( )1

0 1 2 1 22

f ' x x x x= = = = +

Lp bng bin thin ngTnh c:

( ) ( )

( ) ( )

1 2 3 2 2 1 4 2

1 2 3 2 2 1 4 2

= + =

+ = = +

f f

f f

Vy gi trln nht ca P bng 1 4 2+ Du = khi : 1 2, 3 2 2 1 2, y 3 2 2x y z hay x z= = + = = = + =

hoc 3 2 2, 1 2 3 2 2, 1 2x y z hay x z y= = = + = = = +

Bi 28: Cho x, y, z l cc sthc dng.

Tm gi trnhnht ca biu thc3

2 3P

x xy xyz x y z=

+ + + +

Hng dn

Ta c 3 31 1

2 .8 2 .8 .324 8

x xy xyz x x y x y z+ + = + +

( ) ( )2 8 2 8 32 32 48 24 24 3

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

t ( ) 23 2

; 02 3

t x y z t P f t t t

= + + =

( ) ( )3 23 1

; 0 1f t f t tt t

= + = =

Lp bng bin thin ca hm f(t) ta c min3

2P = ti t=1

Du = xy ra khi v chkhi

16

21142 821

2 32 1

21

x

x y z

x y y

x z

z

=

+ + = = = =

=

Bi 29:Cho a, b, c khng m v 2 2 2 3a b c+ + = .

Tm gi trln nht ca biu thc 5a 5 5 4P ab bc ca b c= + + + + + +

Hng dn


17/28



Ta c ( ) ( )2 2 2 23 3a b c a b c + + + +

( )2

3 9a b c + +

3 3a b c + +

t t a b c= + + vi 3; 3t

M ( ) ( )

2 2 2 2 2

32 2

a b c a b c tab bc ca+ + + +

+ + = =

Nn ( ) 21 5

52 2

P t t t = + + . ( )' 5 0, 3; 3P t t t = + > . Lp BBT ta c kt qu.

Vy ax 22mP = vi 3 1t a b c= = = =

Bi 30:Cho cc s thc a, b, c tha mn cba v 5222 =++ cba .

Chng minh rng: 4))()()(( ++ cabcabaccbba

Hng dnTa c: 4))()()(( ++ cabcabaccbba

4))()()(( ++= cabcabcacbbaP

Do cba nn

Nu 0ab bc ca+ + < th 40


18/28



[ ][ ]5;0;36)5()(36)( 3

5;0== xxxxfxfMax

436.9

32 PP

Du "=" xy ra

=

=

=

=++

=

=

=++

=++

=

=

=

01

2

5

2

1

2

5

2

2

222222 c

b

a

cba

ac

ab

cabcab

cba

ca

cbba

x

Bi 31:Cho cc sthc dng x, y, z.

Tm gi trln nht ca biu thc2 2 2

yz xyzxP

x yz y zx z xy= + +

+ + +

Hng dnp dng bt ng thc AM GM ta c

2

1 12 2

yz x x

x y zx yz x yz=

+ ++ + (1)

Tng tta c2

1 12 2

zx y y

x y zy zx y zx=

+ ++ +(2)

21 1

2 2

xy z z

x y zz xy z xy=

+ ++ + (3)

Cng 3 bt ng thc cng chiu (1), (2), (3) ta c 2 2 1P P Du bng xy ra khi x = y = z.

Vy Max P = 1 khi x = y = z.

Bi 32:Cho bn sdng a, b, c, d thomn a + b + c + d = 4

Chng minh rng: a b c d b c c d d a a b2 2 2 2

21 1 1 1

+ + +

+ + + +

Hng dn


19/28



Du "=" xy ra khi v chkhi a = b = c = d = 1.

Bi 33:Cho a,b l hai sthc dng tha 524

a b+ = .

Tm gi trnhnht ca biu thc2 1

4F

a b= +

Hng dn

Ta c :2 1 2 1

8 4 (8 4 )4 4

F a b a ba b a b

= + = + + + + =2 1

8 4 54

a ba b

+ + +

Bt ng thc Csi cho :2

8 8aa

+ v1

4 24

bb

+

Sdng bt ng thc Csi:

2

a ab c ab c ab c ab c ab abca a a a a

b c1+b c b c

2 2

2

(1 )(1)

2 4 4 421

+= = =

+

Du = xy ra khi v chkhi b = c = 1

( )2

bc db bc d bc d bc d bc bcd b b b b b

c d1+c d c d

2 2

2

1

(2)2 4 4 421

+

= = = +

( )2

cd ac cd a cd a cd a cd cdac c c c c

d a1+d a d a

2 2

2

1(3)

2 4 4 421

+= = =

+

( )2

da bd da b da b da b da dabd d d d d

a b1+a b a b

2 2

2

1(4)

2 4 4 421

+= = =

+

T(1), (2), (3), (4) suy ra:

a b c d ab bc cd da abc bcd cda dab

b c c d d a a b2 2 2 2

4

4 41 1 1 1

+ + + + + ++ + +

+ + + +

Mt khc:

( )( ) a c b d

ab bc cd da a c b d

2

42

+ + ++ + + = + + =

.

Du "=" xy ra a+c = b+d

( ) ( ) ( ) ( )a b c d

abc bcd cda dab ab c d cd b a c d b a

2 2

2 2

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

( )( ) ( )( )a b c d

abc bcd cda dab a b c d a b c d

4 4

+ ++ + + + + + = + +

a b c d abc bcd cda dab

2

42

+ + + + + + =

. Du "=" xy ra a = b = c = d = 1.

Vy ta c: a b c d b c c d d a a b2 2 2 2

4 44

4 41 1 1 1+ + +

+ + + +

a b c d

b c c d d a a b2 2 2 2

21 1 1 1

+ + +

+ + + +

pcm.


20/28



Suy ra 5F . 5MinF = t khi

28

11

4 24

15

2 44

, 0

aa

ab

b

ba b

a b

=

= =

=

+ =

>

Bi 34:Cho x,y R v x, y > 1. Tm gi trnhnht ca( ) ( )3 3 2 2

( 1)( 1)

x y x yP

x y

+ +=

Hng dn

t t = x + y ; t > 2. p dng BT 4xy (x + y)2ta c2

4

txy

3 2 (3 2)1

t t xy t P

xy t

=

+. Do 3t - 2 > 0 v

2

4

txy nn ta c

2

3 2 2

2

(3 2)

42

14

t tt t t

Pt t

t

=

+

Xt hm s2 2

2

4( ) ; '( ) ;

2 ( 2)t t t

f t f tt t

= =

f(t) = 0 t = 0 v t = 4.

t 2 4 +f(t) - 0 +

f(t)

+ +

8

Do min P =(2; )min ( )f t

+

= f(4) = 8 t c khi4 2

4 2

x y x

xy y

+ = =

= =

Bi 35:Cho cc sthc dng a,b,c i mt khc nhau tha mn 2a c v 22ab bc c+ = .

Tm gi trln nht ca biu thca b c

Pa b b c c a

= + +

.

Hng dn

Theo githit:1

2 n2

aa c n

c ; 2

22 . 2 1

a b b a cab bc c

c c c c b+ = + = =

V1

2

a

c nn

4

3

b

c . t

ct

b= th

30

4t<

2

2

1 2 1 1 2 71

2 1 1 2(1 ) 2 1 6(1 )1 1

a b

t tc cPa b b a t t t t t t

c c c c

= + + = + + = +

+


21/28



Xt hm s2 7 3

( ) 1 , 0;2 1 6(1 ) 4

f t tt t

= + +

. Ta c:

3'( ) 0, 0;

4f t t

>

, do ( )f t ng bin trn3

0;4

Do GTLN ca hm st ti3

4t= , suy ra

27max

5P =

ng thc xy ra khi22

8 3 42

ab bc ca b c

a c

+ = = =

=, chng hn chn c (a,b,c)=(3,8,6).

Bi 36: Cho , ,a b c l cc sdng v 3a b c+ + = .


bc ca ab

a bc b ca c abP + +

+ + +=

Hng dn

V a + b + c = 3 ta c3 ( ) ( )( )

bc bc bc


+ + + + + +

1 1

2

bc

a b a c

+

+ +


( )( )a b a c a b a c+ + + + + , du ng thc xy ra b = c

Tng t1 1

23

ca ca

b a b cb ca

+

+ ++ v

1 1

23

ab ab

c a c bc ab

+

+ ++

Suy ra P3

2( ) 2( ) 2( ) 2 2


a b c a b c

+ + + + + + + = =

+ + +,


2khi a = b = c = 1.

Bi 37: Cho hai sdng x, y thomn iu kin x + y = 4. Tm gi trnhnht ca biu

thc331 11 1 = + + + + +

S x y

x y

Hng dnTheo bt ng thc Csi cho ba sdng ta c:

3 3 31 7 7 7 7 1

1 3. . 1 (1)2 2 2 2

+ + + + + +

x x

x x

3 3 31 7 7 7 7 1

1 3. . 1 (2)2 2 2 2

+ + + + + +

y y

y y

Cng tng vca (1), (2) ta c33 231 1 7 7 1 11 1 3. 2

2 2

+ + + + + + + + + +

x y x y

x y x y

Mt khc ta li c ( )1 1 1 1 1 4

4 . 4

+ + = + +

x y xyx y x y x yxy

nn

33 231 1 7 7 41 1 3. 2

2 2

+ + + + + + + + +

+ x y x y

x y x y


22/28



Theo githit x = y = 4 nn237 7 343

3. .72 2 4

+

S S

Du = xy ra khi v chkhi

1 71

21 7

1 22

4

+ + =

+ + = = =

=

+ =

xx

y x yy

x y

x y

Vy343

min4

=S

Bi 38: Xt cc sthc dng x, y, z tha mn iu kin x + y + z = 1.

Tm gi trnhnht ca biu thc:2 2 2x (y z) y (z x) z (x y)

Pyz zx xy

+ + += + + .

H

ng dn

Ta c :2 2 2 2 2 2x x y y z z

Py z z x x y

= + + + + + (*)

Nhn thy : x2+ y2 xy xy x, y R

Do : x3+ y3xy(x + y) x, y > 0 hay2 2x y

x yy x

+ + x, y > 0

Tng t, ta c :2 2y z

y zz y

+ + y, z > 0

2 2z xz x

x z+ + x, z > 0

Cng tng vba bt ng thc va nhn c trn, kt hp vi (*), ta c:P 2(x + y + z) = 2 x, y, z > 0 v x + y + z = 1

Hn na, ta li c P = 2 khi x = y = z = 13

. V vy, minP = 2.

Bi 39: 0, 0x y> > tha mn 2 2 3x y xy x y xy+ = + + . Tm gi trnhnht ca biu thc2

2 2 (1 2 ) 32xy

P x yxy

+ = + +

Hng dn


23/28



Hay gi trnhnht ca P bng71

4khi x = y = 2

Bi 40: Chox, yl hai sthc dng tha mn 2 3 7x y+ . Tm gi trnhnht ca biu

thc 2 2 2 232 5( ) 24 8( ) ( 3)P xy y x y x y x y= + + + + + + .

Hng dn

Ta c

22 2 3 3

6( 1)( 1) (2 2)(3 3) 36 52

x y

x y x y x y xy

+ + + + + = + + + +

.

Ta c ( )22 2 2 25( ) 2 5( ) 2x y x y x y x y+ + + + v

2 2 2

2 2

( 3) 9 2 6 6 0

2( 3) 8( ) ( 3)

x y x y xy x y

x y xy x y x y

+ = + + +

+ + + + + +

Suy ra 32( ) 24 2( 3)P xy x y x y xy + + + + +

t ( ], 0;5t x y xy t = + + , 3( ) 2 24 2 6P f t t t = +

Ta c ( ]23

/

2 23 3

(2 6) 824.2( ) 2 2 0, 0;5

3 (2 6) (2 6)

tf t t

t t

+ = = <

+ + Vy hm sf(t) nghch bin trn na khong ( ]0;5 .

Suy ra 3min ( ) (5) 10 48 2= = f t f . Vy 32

min 10 48 2,1

xP khi

y

==

= Bi 41:Xt cc sthc khng m x, y, z thomn iu kin: 2 2 2 3x y z+ + = . Tm gi

trln nht ca biu thc P xy yz zx= + + +4

x y z+ +

+ Ta c :2 2 3

( ) 3 (1) 0, 0 0

x y xy x y xy

xy x y x y xy do x y n n x y

+ = + +

+ = + + > > + >

[ ][ ]

2

2 2

1 1 4(1) 3 3 ( ) 3( ) 4 0

( ) 1 ( ) 4 0 41 3 3 1

(1) 1 1

1 3 ( ) 2 ( ) 1

x y x y x yx y x y

x y x y x y

xy x y x y xy

N n P x y x yxy x y

+ = + + + + + +

+ + + +

= + =+ +

= + + = + + ++

+t 23

( 4) 1 ( )x y t t P t f tt

+ = = + + =

+ Ta c3

2 2

3 2 3'( ) 2 0, 4

tf t t t

t t

= = > > Nn f(t) ng bin trn

[ ) 714; ( ) (4)4

P f t f + = =


24/28



Hng dn

( ) ( ) ( )

( )

22 2 2 2

2

31

2 2

3 4

2

+ + + + + + =

+ + +

+ +

x y zx y z x y z

x y z

x y z

Ta co: xy + yz + zx =

Do o P=

( )( )

( )

2 2 2

22

2

3

33 0 3 6

2

3 9. 3 3

+ + =

+ + + +

+ + + +

x y z

x y zx y z

x y z x y z

V 0 xy + yz + zx

Nen 0

Suy ra

( )

2

2 3

2 2

3 3

3 43 3,

2

3 4 4 43 3,

2' 0 4 4

+

+ =

= = =

tt

t

t tt

t t t

f t t t

at t =x+y+z, P=

Xet f(t)= vi f'(t)= t-

(loai)

( ) ( )

( )

4 3 133 , 3

3 313 13

3 3,3 3

13 13.

3 3

= =

f f

t tNen f khi do o P

Khi x=y=z=1 th P= Do o gia tr ln nhat cua P la

Bi 42: Tm gi trnhnht, gi trln nht ca hm s2 2 2( ) 5 8 32 3 24 3 12 16f x x x x x x x= + + + + .

Hng dnTa c TX: [0;8]D =

t : 2 2( ) 5 8 32, ( ) 3 12 16g x x x h x x x= + = + Ta ddng xc nh c [0;8]x , th

6 (2) ( ) (8) 12 2, 2 (2) ( ) (8) 4 7g g x g h h x h= = = =

v 2 20

3 24 0 ( 3 24 0 )

8

xx x x x

x

= + + =

=

.

Do 2

2

2 2

8( 2)( ) 3 12 16 0 ( ) 2 [0;8]

5 8 32 3 24

xf x x x h x x

x x x x

= + + +

+ + +

.

ng thc xy khi v chkhi x = 2 min ( ) 2f x = khi x= 2.Ta c

2 2 2( ) 5 8 32 3 24 3 12 16 ( ) ( ) 12 2 4 7 [0;8].f x x x x x x x g x h x x= + + + + + +

ng thc xy khi v chkhi x = 8 m ax ( ) 12 2 4 7f x = + khi x= 8.


25/28



Vy min ( ) 2f x = khi x= 2 v m ax ( ) 12 2 4 7f x = + khi x= 8.

Bi 43: Cho cc sthc khng m x, y tha mn 2 2 (3 2)( 1) 0x y x y+ + =

Tm gi trln nht ca biu thc 2 2 8 4P x y x y x y= + + + +

Hng dn+Ta c 2 2 2(3 2)( 1) 0 ( ) 3( ) 2x y x y x y x y xy y+ + = + + + =

V x,y khng m nn2

( ) 3( ) 2 0 1 2x y x y x y+ + + + t t = x+y khi [ ]1;2t

Ta c 2 2 28 4 ( ) ( ) 8 4 ( )P x y x y x y x y x y x y= + + + + + + + + + 2 8 4P t t t + +

+Xt hm 2( ) 8 4f t t t t= + + vi [ ]1;2t

ta c4

'( ) 2 14

f t tt

= +

vi [ ]1;2t4

'( ) 3 02

f t > > vi [ ]1;2t

v f(t) lin tc trn on [1;2] nn f(t) ng bin trn on [1;2]

[1;2]

( ) (2) 6 8 2 ( ) 6 8 2maxf t f f t = = + +

6 8 2P + , P= 6 8 2+ khi. 0

2

x y

t

=

=

2

0

x

y

=

=

KL: Gi trln nht ca P l 6 8 2+ t c khi x = 2 v y = 0

Bi 44: Cho a,b,c l cc s thc khng m v tha mn 3a b c+ + = . Tm gi tr ln

nht ca biu thc: 3 2 2 2 2 2 22( ) 27 3( ) 6( )P ab bc ca a b c a b c ab bc ca= + + + + + + + +

Hng dn

Ta c: 33 . .ab bc ca ab bc ca+ + 2 2 2 327 ( )a b c ab bc ca + +

Li c: 2 2 2 2 2 23( ) 3( )a b c ab bc ca a b c ab bc ca+ + + + + + + + Do 3 3( ) 3( ) 3 ( )P ab bc ca ab bc ca t t f t + + + + + = + =

vi2( )

0 13

a b ct ab bc ca

+ + = + + =

Ta c bng bt ca hm sf(t)trn [ ]0;1

TBBT ta c:[ ]0;1ax ( ) 2

tM f t

= khi t=1

T ta c GTLN caP bng 2 khi1

3a b c= = =

t 0 1

f(t) + 0

f(t)

0

2


26/28



Bi 45: Cho a, b, cl cc sthc thomn 3.a b c+ + =

Tm gi trnhnht ca biu thc 4 9 16 9 16 4 16 4 9 .a b c a b c a b cM = + + + + + + + +

Hng dn

+ t ( ) ( ) ( )2 ;3 ;4 , 2 ;3 ;4 , w 2 ;3 ;4 wa b c c a b b c au v M u v= = = = + +

( ) ( ) ( )2 2 2w 2 2 2 3 3 3 4 4 4a b c a b c a b cM u v + + = + + + + + + + +

+ Theo c si c 322 2 2 3 2 6b c a b c+ ++ + = . Tng t+ Vy 3 29.M Du bng xy ra khi 1.a b c= = = Bi 46: Cho cc sthc dng , ,x y z tha mn 3x y z+ + = .

Tm gi tr nh nht ca biu thc 2 2 22 2 2

xy yz zxP x y z

x y y z z x

+ += + + +

+ +.

Hng dn

p dng BT Csi ta c3 2 2

2x xy x y+ . Tng t3 2 2 3 2 2

2 , 2y yz y z z zx z x+ + ( ) ( ) ( )3 3 3 2 2 2 2 2 22 1x y z x y y z z x xy yz zx + + + + + + . Mt khc, do 3x y z+ + = nn

( ) ( )( )2 2 2 2 2 23 x y z x y z x y z+ + = + + + +

( ) ( )( )3 3 3 2 2 2 2 2 2 2x y z x y y z z x xy yz zx= + + + + + + + +

T(1) v (2), ta c 2 2 2 2 2 22 2 2 2 2 2

1 1x y z x y y z z x

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

+ + + +(3)

Ta c ( ) ( ) ( )2 2 2 2 2 2 22 9 2x y z x y z xy yz zx x y z xy yz zx+ + = + + + + + = + + + + + .

t 2 2 29

2

tt x y z xy yz zx

= + + + + = . Do ( )

22 2 23( ) 3 9 3x y z x y z t t+ + + +

T(3) ta c 2 2 22 2 2

xy yz zxP x y z

x y z

+ + + + +

+ +. Khi

29 2 9

2 2

t t tP t P

t t

+ + vi 3t

Xt hm s ( )22 9

2

t tf t

t

+= trn [ )3;+ .

C2 2

2 2

(4 1) (2 9) 2 9'( ) 0, 3

2 2

t t t t t f t t

t t

+ = = > . Lp BBT, ta c :

( )3

( ) min 4t

P f t f t

= . Du bng xy ra khi 3 1t x y z= = = =

Vy min 4 1P x y z= = = =

Bi 47: Cho x , y , z l ba sthc tha mn : 2x + 3y + z = 40.

Tm gi trnhnht ca biu thc: 2 2 22 1 3 16 36S x y z= + + + + +

Hng dn

- Ta c: ( ) ( )2 22 2 2 22 2 3 12 6S x y z= + + + + + .

- Trong htoOxy xt 3 vc tsau:

( ) ( ) ( )2 ;2 , 3 ;4 , ;6a x b y c z= = =

, ( ) ( )2 3 ;2 12 6 40;20a b c x y z+ + = + + + + =


27/28



( ) ( ) ( )2 2 22 2 22 2 , 3 12 , 6a x b y c z= + = + = +

, 20 5a b c+ + =

Sdng bt ng thc vdi vc t:

S=a b c a b c+ + + +

20 5S .

ng thc xy ra khi cc vc t , ,a b c

cng hng

2 3 2 3 2 3 40

22 12 6 2 12 6 20 20

x y z x y z x y z+ +

= = = = = = = 2, 8, 12x y z = = = Vi : 2, 8, 12x y z= = = th 20 5S =

Vy gi trnhnht ca S bng 20 5 t c khi : 2, 8, 12x y z= = =

Bi 48:Gisx,y,zl cc sthc dng tha mn 2 2 2 1.x y z+ + =

Tm gi trln nht ca biu thc3 3 3 3

2 2 3 3.

1 1 24

xy yz x y y zP

z x x z

+= +

+ +

Hng dn

p dng bt ng thc Csi ta c :

( ) ( ) ( ) ( )

2 2 2 2 2 2 2 2 2 21 1

xy yz xy yz

z x z x z y x y x z

+ = ++ + + + + + + +

( )( ) ( )( )

2 2 2 2

2 2 2 2 2 2 2 22 2 2 2 2 2 2 2

2 2 2 2

2 2 2 2

1

42 2

1 1 1 1 11 1 1 .

4 4 2 2 4 2 2 4 8

xy yz x y y z

z x z y x y x zz x z y x y x z

y y y y y y y y

yz xy z x z xz y x y

+ + + +

+ + + + + + + +

= + + + + = + + = + +

+ +

Tip tc p dng bt ng thc C si, ta c 3 3 3 3 31

( )4

x y y z xy yz+ +

nn :33 3 3 3 3

3 3 3 3

( ) 1.

44

x y y z xy yz y y

z xz x z x

+ + = +

Suy ra

31 1 1

.4 8 96

y y y yP

z x z x

+ + +

t ,y ytz x

= + khi 0t> v 31 1 1 .96 8 4

P t t + +

Xt hm s 31 1 1

( )96 8 4

f t t t= + + vi 0.t>

Ta c 21 1

'( ) ; '( ) 0 2,32 8

f t t f t t= + = = v 0.t>

Suy ra bng bin thin:

Da vo bng bin thin ta c5

,12

P du ng thc xy ra khi v ch khi 2t= hay

1.

3

x y z= = = Vy gi trln nht ca Pl5

,

12

t c khi1

.

3

x y z= = =

Bi 49: Cho , ,a b c l cc sthc dng thomn 7ab bc ca abc+ + = .

Tm gi trnhnht ca biu thc :4 5 6

2 2 2

8 1 108 1 16 1a b cS

a b c

+ + += + + .

Hng dn

( )f t

'( )f t

t 2

0+

0 +

5

12


28/28


Githit tng ng vi1 1 1

7a b c

+ + = p dng bt ng thc Csi+Bunhiacpxki ta

c: 2 3 32 2 2 2

1 2 2 28 54 54

2 9 9 9S a b b

a b b b

= + + + + + + +

42 2

1 116

4 4c

c c

+ +

22

2 2 2

1 1 1 1 1 1 1 14 10 3 17 .7 24

2 3 2 2 3 2 7a b c a b c

+ + + + + + + + = + =

+ +

du bng xy ra khi

1 1,

2 3a c b= = = .Vy gi trnhnht ca Sbng 24 t khi

1 1,

2 3a c b= = =

Bi 50: Cho a, b, c l cc sdng tha mn a b c 1+ + = .Chng minh rng: a b b c c a 3

ab c bc a ac b

+ + ++ +

+ + +

Hng dn

Ta ca b 1 c 1 c

ab c ab 1 b a (1 a)(1 b)

+ = =

+ +

VT=1 c 1 b 1 a

(1 a)(1 b) (1 c)(1 a) (1 c)(1 b)

+ +

Do a,b,c dng v a+b+c =1 nn a, b, c ( )0;1 1 a;1 b;1 c dngp dng bt C si cho 3 sdng ta c

VT1 c 1 b 1 a

33 . . 3(1 a)(1 b) (1 c)(1 a) (1 c)(1 b)

=

(pcm)

ng thc xy ra khi1

a b c3

= = =

Fb: https://www.facebook.com/ng.huubien

Documents

Tuyển Chọn 50 Bài Toán Điển Hình Min-max