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Hunh Ch Ho

Gi s ta cn chng minh bt ng thc ba bin dng:

( ), , 0f x y z

vi , ,x y z l cc bin s thc tha mn cc tnh cht no .

Khi ta s thc hin hai bc chnh sau y:

Bc 1: Chng minh ( ) ( ), , , ,f x y z f t t z

i vi bt ng thc khng iu kin th dn bin theo cc i lng trung bnh:

2 2

, ;2 2x y x y

t t xy t+ +

= = = ,

Bc 2: Chng minh ( ), , 0f t t z

Kt lun: ( ), , 0f x y z

Ch : i vi cc bt ng thc ng bc ta c th lm cho chng n gin hn bng cch chun ha cc bin trong bt ng thc trc khi thc hin hai bc.

Th d 1. Cho , , 0x y z . Chng minh rng:

33x y z xyz+ + (1)

Li gii.

CCH 1: Thc hin dn bin theo TBC Bc 1:

Ta c: ( ) 31 3 0x y z xyz + + (2) Xt biu thc ( ) 3, , 3f x y z x y z xyx= + + . Ta chng minh: ( ), , 0f x y z Thc hin dn bin theo TBC:

2x y

t+

= , ta s chng minh:

( ) ( ), , , ,f x y z f t t z (3) Tht vy, xt hiu: ( ) ( ), , , ,d f x y z f t t z=

2333 2 3x y z xyz t z t z = + + +

( )23 33x y t z xyz= + + M

2x y

t+

= 2t xy 23 3 0t z xyz nn 0d

Bc 2: Chng minh ( ) 23, , 2 3 0f t t z t z t z= + (4) Tht vy: ( ) ( )32 234 2 3 2 27 0t z t z t z t z + +

( ) ( )2 8 0t z t z + (ng) Kt lun: ( ), , 0f x y z

CCH 2: Thc hin dn bin theo TBN

Bc 1: Ta c: ( ) 31 3 0x y z xyz + + (2) Xt biu thc ( ) 3, , 3f x y z x y z xyx= + + . Ta chng minh: ( ), , 0f x y z Thc hin dn bin theo TBN: t xy= , ta s chng minh: ( ) ( ), , , ,f x y z f t t z (3) Tht vy, xt hiu: ( ) ( ), , , ,d f x y z f t t z=

2333 2 3x y z xyz t z t z = + + +

2x y t= +

M t xy= 2t x y + 2 0x y t + nn 0d

Bc 2: Chng minh ( ) 23, , 2 3 0f t t z t z t z= + (4) Tht vy: ( ) ( )32 234 2 3 2 27 0t z t z t z t z + +

( ) ( )2 8 0t z t z + (ng) Kt lun: ( ), , 0f x y z

CCH 3: Chun ha & thc hin dn bin theo TBC

V bt ng thc (1) l ng bc nn bng cch chun ha ta c th gi s: 1x y z+ + = (*) Bc 1:

Ta c: ( ) 31 1 3 1 27 0xyz xyz (2) Xt biu thc ( ), , 1 27f x y z xyz= . Ta chng minh: ( ), , 0f x y z Thc hin dn bin theo TBC:

2x y

t+

= , ta s chng minh:

( ) ( ), , , ,f x y z f t t z (3) Kim tra (*): Khi thay ,x y bi

2x y

t+

= th (*) vn tha

Xt hiu: ( ) ( ), , , ,d f x y z f t t z= ( )21 27 1 27xyz t z= ( )227 t z xyz= M

2x y

t+

= 2t xy 2xyz t z nn 0d

Bc 2: Chng minh ( ) 2, , 1 27 0f t t z t z= (4) Tht vy: ( ) ( ) ( )( )2 2 2, , 1 27 1 27 1 2 1 6 1 3 0f t t z t z t t t t= = = + Vi iu kin (*) th ng thc xy ra 1

33 1x y

x y zt

= = = =

=

Vy trong trng hp tng qut ng thc xy ra 0x y z= = Kt lun: ( ), , 0f x y z

CCH 4: Chun ha & thc hin dn bin theo TBN

V bt ng thc (1) l ng bc nn bng cch chun ha ta c th gi s: 1xyz = (*) Bc 1:

Ta c: ( )1 3 3 0x y z x y z + + + + (2) Xt biu thc ( ), , 3f x y z x y z= + + . Ta chng minh: ( ), , 0f x y z Thc hin dn bin theo TBC: t xy= , ta s chng minh: ( ) ( ), , , ,f x y z f t t z (3) Kim tra (*): Khi thay ,x y bi t xy= th (*) vn tha Xt hiu: ( ) ( ), , , ,d f x y z f t t z= ( )3 2 3x y z t z= + + + 2x y t= +

M t xy= 2 2x y xy t+ = 2 0x y t + nn 0d Bc 2: Chng minh ( ), , 0f t t z (4)

Tht vy: ( ) ( ) ( )2

2 2

1 2 11, , 2 3 2 3 0

t tf t t z t z t

t t

+= + = + =

Vi iu kin (*) th ng thc xy ra 1 11

x yx y x y z

t

= = = = = =

=

Vy trong trng hp tng qut ng thc xy ra 0x y z= = Kt lun: ( ), , 0f x y z

Th d 2. Cho , ,a b c l cc s thc dng tha mn iu kin . . 1a b c = . Chng minh rng:

1 1 1 13 251 4

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

(1)

Li gii.

Xt biu thc ( ) 1 1 1 13, ,1

f a b ca b c a b c

= + + ++ + +

.

Thc hin dn bin theo TBN, ta s chng minh: ( ) ( ), , , ,f a b c f a bc bc (3) Ta c: ( ) ( ), , , ,d f a b c f a bc bc= 1 1 1 13 1 2 131 2 1a b c a b c a bc a bc = + + + + + + + + + +

1 1 2 1 113

1 2 1b c a b cbc a bc

= + +

+ + + + +

( ) ( )( )2 1 13

1 2 1b c

bc a b c a bc

= + + + + +

Khng mt tnh tng qut, ta gi s { }max , ,a a b c= , do 1abc = 1bc 1 1bc

Mt khc, theo bt ng thc AM-GM ta c:

( )( ) ( )( )3 313 13 13

1161 2 1 3 1 3 1a b c a bc abc abc

=