PHNG PHP DN BIN(Mixing variable)TRONG CC K THI !"I H#C NH$NG N%M
G&N !'()* !+T ,-N !.K/ 01 n23 456 7889:7887; ?i 456 >@A6 0B
64C6 D3 chungE; vF vGH6I6 bJi 0KIn bL0 >Mng 04C6 N0 xOL0 4iPn vJ
nQO 6R 04F >S 4R6 bTa 6Ung giV3 >i*MW6 XY vGH; 64K XY N0 xOL0
4iPn n4@ng 6I6 bJi 0KIn bL0 >Mng 04C6 0rKng 6I6 ?i 456 6Ung Mng
04C6 >\i xCng; 4KIn v] ^DK >R nQO vGn X_ng lin4 4K?0 `4@ang
`4I` Xbn biQn 6I6 bJi 0KIn bL0 >Mng04C6 0rc ndn Xe XJng 4an
TrKng ng4idn 6CO 6fa 0Zi; N64 6fa 0Zi lJ gh X_ng 3S0`4@ang `4I`
64Ong; `4@ang `4I` Xbn biQn 64K 0L0 6V 6I6 bJi 0KIn bL0 >Mng
04C60rKng 6I6 ?i 456 3J 64Tng 0Zi >l 6G` >Qn*!l 0Ji ng4idn
6CO gi giT` giIK vidn vJ 456 gin4 6R 3S0 0Ji liPO 0iQ` 6Gn
vmi`4@ang `4I` Xbn biQn; 3S0 `4@ang `4I` 3mi >/ giVi 0\0 6I6 bJi
0KIn bL0 >Mng04C6 * DK 04ni gian vJ l 0Ji Mng 04C6 6a bVnB-T
!xNG THyC C{ kITrKng0KJnbSng4idn6CO;
64Tng0Zi3Kng3O\n>@abJi0KInn4ilObiQnvln4ilO n4L0 lJ 4ai biQn; XK
>R 64Tng 0Zi 64| `4I0 bi/O bL0 >Mng 04C6 CZ:gi 64| cX?ng 6a
bVn n4L0*Gi s , x yl 2 s thc khng m. Khi 2x yxy+ng thc xy ra khi
xy,J gaO >tH lJ 6I6 >]n4 lq 6a bVn n4L0 0rKnggI64 giIK RlJ
6Zng 6_ bB 0rA 04iQ0 04}6 64K giVi 0KIn bL0 >Mng 04C6*b* CI6
>]n4 lq 6~n 04iQ0!NH 9!h" hm s( )y f x =li#n t$c tr#n "%n
a&'. Khi t(n t%i gi) tr* l+n nh,t -gi) tr* nh. nh,t c/a hm s
tr#n "%n .!NH 7!h" hm s( )y f x =li#n t$c tr#n "%n a&' - %" hm(
)' 0, ; f x x a b " .Khi ( ) ( ) ( )f a f x f b !NH !h" hm s( )y f
x =li#n t$c tr#n "%n a&' - %" hm( )' 0, ; f x x a b " .Khi ( )
( ) ( )f b f x f a 6* PHNG PHP DN BINTrKng 6I6 ?i 456; nQO 6R bJi
0KIn bL0 >Mng 04C6 04F 6Ung Mng 04C6 >\i xCng 4aH 4KIn v]* MJ
nQO 04Q 04F gh X_ng `4@ang `4I` XbnbiQn gaO >tH gi 6}6 @A6 W0
tlJ 6I6 >?i l@Ang 0rOng bFn4 >/ i 6I60Nn4 64L0 64K 0r@m6;
64Mng 4?n 2 2, , ,....2 2x y x yt t xy t+ += = =Bc 2* !In4 giI( ),
, 0 f x t t *P4@ang `4I` 64Tng 0Zi >l 6G`>Qn64|ngjn ng5n n4@
04Q;viP6 In4 giI( ) ( ), , , , f x y z f x t t * !ilO >R gh X_ng
n4ilO an giVn vF 64Tng 0a > 4?n 64Q 6n l?i 64| 7 biQn g\*T4N X_
( BL0 >Mng 04C6 CZgi 64K g\)Gi s , , x y zl 2 s thc khng m. Khi
33 x y z xyz ++ .ng thc xy ra khi xy1C4Cng 3in4B@m6 9!W0 ( )3, , 3
f x y z x y z xyz =++-!W0 2x yt+= ; 0a 6R 2t xy gOH ra( )( )3 23, ,
( , , ) 3 0 f x y z f t t z t z xyz - = - B@m6 7 lJ 64Cng 3in43 2(
, , ) 2 3 0 f t t z t z t z = +- ,iP6 nJH 04G0 >an giVn vF( ) (
) ( )3 23 2 22 3 0 2 27 0 8 0 t z t z t z t z t z t z +- + - - +
,GH( ) ( ), , , , 0 f x y z f t t z !Mng 04C6 xVH ra Mng 04C6 CZgi
64K g\ X@ang v1a >@A6 64Cng 3in4; 64Tng 0Zi gi gh X_ngnR n4@ lJ
3S0 bB >l 64K 6I6 64Cng 3in4 0iQ` 04eK*7* CC BI TON B-T !xNG
THyC , CCH GII CHwNGP4~n nJH 64Tng 0Zi 64| gh X_ng bL0 >Mng 04C6
CZgi 64K 7 g\; gh X_ng 0Nn4 >an>iPO 6fa 4J3 g\; vJ 6I6 biQn
>Bi 04Zng 04@nng >/ 64Cng 3in4 6I6 bL0 >Mng 04C6*Ng4a lJ
64Tng 0Zi 3O\n 64| 6~n 6I6 ?i g\ vJ 6I6 bL0 >Mng 04C6 l@Ang
giI6* a* CI6 bL0 >Mng 04C6 >?i g\9* KHI ): 7899 Cho x, y, z l
3 s thc thuc 1;v ; x y x z . Tm gi tr nh nhtc!" #i$u th%c 2 3x y
zPx y y z z x= + ++ + +&i'i(Ta 6R 1 1 12 32 3 1 1x y zPy z xx y
y z z xx y zx y zx y z= + + = + +! " ! " ! "+ + +# # #$ $ $+ + + #
# #$ $ $# # #$ $ $# # # $ $ $% & % & % &!W0, ,y z xa b
cx y z= = =; bJi 0KIn 0rc 04Jn4 0F3 giI 0r] n4o n4L0 6fa1 1 12 3 1
1Pa b c= + ++ + + vmi >ilO ilO Tng vF 1 t vJ 2 0 b c bc +- * DLO
b[ng gVH ra R( )( , , ) , , 0 f a bc f a t t - 4aH( )( , , ) , , f
a bc f a t t B@m6 6n l?i 6fa `4@ang `4I` Xbn biQn >In4 giI ( ),
, f a t t*T4eK giV 04iQ0 21. . 1 abc at= =ndn22 21 2( , , ) , ,1 2
3tf a t t f t tt t t! "#$# = = +$#$# $+ + % &!W0 ( ) ( )( ) (
)( ) ( )322 2223 3 2 1 '2' 01 2 32 3 1t t t ttf t f tt tt t - - + -
+ = + = (+ ++ +kOH ra( ) ( )3233f t f = * DLO b[ng gVH ra an giVn
4an* 7*KHI ): 788Cho x, y, zl 3 s thc )*ng th"+ ,-i th" m.n( )3 x x
y z yz ++ =.Ch%ng minh r/ng ( ) ( ) ( ) ( ) ( ) ( )3 3 33 ) x z x y
x y x zy z y z + + + + + + + +Ch%ng minh( !tH lJ 3S0 >l 0KIn rL0
g~n >tH; tH lJ bJi 0KIn Xe;nR>@A6 >Zng >VK ng@ni 3d
0KIn bL0 >Mng 04C6 gZi nBi >@a ra 6I6 >I` In tH lJ >I`
In >@A6 >@a ra 6fa bS giIK X_6 z >JK 0?K vJ 6I6 >I` In
>@A6 >@ara 6I6 b?n 6R 04/ 0F3 >@A6 0rdn 3?ng; 6O\i lJ
>I` In 6fa 64Tng 0Zi* 01 n c!" # &230T!W0 ax3y0 'x310 cy31 ;
>ilO Mng 04C6 6~n 64Cng 3in4 0@ang >@ang vmi 3 3 33 ) a b abc
c + + T4oa 3n >ilO R gOH ra >`63; XLO b[ng xVH ra tH lJ
`4@ang `4I` 6fa 64Tng 0Zi* 4h*ng 1h1 )5n #i6n!W0( ), , f x y z VT
VP = -Oan gI0bL0>Mng 04C6 6~n 64Cng 3in4 0a 04LH r[ng 6R 4ai
biQn 6R vai 0r n4@n4aO lJ y vJ 1; vJ 04LH >@A6 XLO b[ng xVH ra
W0 2y zt+=K4i >R 04eK giV 04iQ0 ( ) ( )2232 3 3
x x t yz y z t + = + =,F vGH 2 2 22x tx t t + + 4aH( ) ( )2 22 x
t t + * kOH ra( )1 x t Ta 6R( ) ( ) ( ) ( )3 2 3, , 2 * ) f x t t x
t t x t y z = + + + - +( ) ( ) ( ) ( ) ( ) ( )3 3 3 2, , , , 2 * f
x y z f x t t x z x y x t t y z - = + + + - + - -s0 4J3 g\ ( ) ( )
( ) ( ) ( )3 3 3 22 * , 0; g x x z x y x t t y z x t= + + + - + - -
Ta 6R( ) ( ) ( ) ( )2 2 2' 3 2 g x x z x y x t = + + + - + ( ) ( )
( ) ( )'' * 2 0 gx x z x y x t = + + + - + = ,GH ( ) ( )23' '(0)
02g x g y z = = - DK >R 04eK >]n4 lq 7 0a 6R( ) ( ) ( ) ( ) (
) ( )3 3 3 22 2 * g x g t t y t z t t y z = + + + - - -( )( ) 33 2
2 333 2 2 33 33 3t y t t y ty yt z t t z tz z+ = + + ++ = + + +( )
( )3 33 2 2 2 3 33 2 2 2 23 2 22 3 ( ) 3( )8 (3 3 2 2 2)8 () ) 2)t
y t z t t y z t y z y zt t y z y z yzt t y z yz+ + + = + + + + + +=
+ + + + -= + + -Ndn ( ) ( )2 22 2 2( ) ) ) 2 2( ) * 3 0 gt t y z yz
y z y z t y z! "#$ = + - - + - - =- - #$ #% &DK >R ( ) ( ) (
) ( ), , , , 0 , , , , f x y z f x t t f x y z f x t t - P4@ang
`4I` Xbn biQn 64| 6n l?i viP6 >In4 giI bi/O 04C6 4ai biQn g\( )
( ) ( )3 23, , 2 * 0 0 f x t t x t t x t t = + + + - 0rKng >ilO
an giVn vF ( ) ( ) ( )3 23, , 2 * 0 0 f x t t t t t t t t + + + -
=,GH tH 64Tng 0Zi v1a >@a ra 3S0 bL0 >Mng 04C6 W6 0r@ng 64K
`4@ang `4I` 64Tng 0Zi >l 6G` >Qn 5h6n x7t8 ,iP6 In4 giI( ) (
), , , , f x y z f x t t ** !l X} b]Mng 04C6 CZgi 04F 2 2
1 1 2 3 2 1 23 2 1 2 2 t t ttt t t t+ =+ + + +=+ + + kOH ra(
)82
81 232, , 3 2f a t t ttttt t+ + +,+ -Tmi >tH; 64Tng 0a 6R 04/
gh X_ng `4@ang `4I` 4J3 g\ >/ 0F3 GTNN 6fa( ), , f a t t; 4KW6
6R 04/ gh X_ng BL0 >Mng 04C6 CZgi 64K g\ 8 8
2,2 ,2 t tt,J `63*DLO b[ng xVH ra ilO giV gh x t ndn221( , , ) 3
f x t t tt +B[ng 6I64
