-

1/2011 1. : n\ ( ) ( ) ( )2 2 21 1 2 2 1 2 3 1: ... , : ... , : max ,n n i n ix x x x x x = + + + = + + + =x x x x . ( )1 2, ,..., nnx x x= x \ , ( )1 2, ,..., nnx x x= x \ , :

( ) ( ) ( ) ( )3 1 2 3n x x xn

x

. . \

, : ( )1 2 3, ,..., nx x x=x \

( ) ( ) ( )22 2 2 2 2 23 1 1 2 1 21 1max ... max ... ,i n ii n i n nx x x x x x x x + + + + + +x x

. : ( ) ( )

( )2 2 2

1 2 1 2 1 2

22 2 21 2 1 2

2 2 2 2 2 21 2 1 2

1

1

... ...

... ...

... ... 2

0 2 , .

n n

n n

n ni j n

j ji j n

x x x x x x

x x x x x x

j jx x x x x x x

x x <

<

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

x x

x

:

( ) ( ) ( )2 3 1 2 1... max , .n ii nn x x x n x + + + x x ( ) ( ) ( ) ( )3 1 2 3n x x x

( )x , ,

. ,

nx \

( )1 2, x x n\0

,

, ,

> , ( ) ( ) (1 )x1 2a x x , . nx \

2. .

1dn\

( ) ( )( )1 1,

, :1 ,

dd

d= +

x yx y

x y,

. n\ : d

2

. ( ) ( )( )1 1,

, : 01 ,

dd

d= +

x yx y

x y, , nx,y \ ( )1 , 0d x y , , nx y \ .

:

( ) ( )( ) ( )1 11,

, : 0 ,1 ,

dd d

d= = +

x yx ,=y x y x y

x y

. 1dn\

.

( ) ( ) ( )( )( )( ) ( ) ( )1 1 1 11 1

, ,, , , , ,

1 , 1 ,d d

d d d dd d

= = =+ +x y y x

x y y x x y y xx y y x

n x,y \ . III.

( ) ( ) ( ) ( )( )( )( )

( )( )

( ) ( )( ) ( )( ) ( ) ( )( ) ( )( )( ) ( )( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1

1 1 1

1 1 1 1 1 1

1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

, , ,, , ,

1 , 1 , 1 ,

, 1 , 1 , , 1 , 1 ,

, 1 , 1 ,

, , , , , , , ,

, , , , , , 2 ,

d d dd d d z y

d d d

d d d d d d

d d d

d d d d d d d d

d d d d d d d d

+ ++ + + + + + +

+ + + + + + + + + +

x y x z z yx y x z

x y x z z y

x y x z z y x z x y z y

z y x y x z

x y x y x z x y z y x y x z z y

x z z y x y x z x y z y x z ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1 1 1

1 1 1 1 1 1 1 1

,

, , 2 , , ,

, , , 2 , , 2 , , ,

d d d d d

d d d d d d d d

+ + + + +

z y

x y z y x y x z z y

x ,y x z z y x z z y x y x z z y

. 3. , :

( )2

11 3 1 1, 1 , 1,2,3,..., , , , 1, 2,3,...2 1

nn

n n

nn n

n n n nn

= = = = + x y

: 1lim 0n n+

= 33lim 1n

ne

n

+ = , ,

, :

( )31 3 1 3lim lim , 1 lim , lim 1 0,n nnn n n n en nn n + + + + = = =

x .

( )1lim2 1

n

n

nn+

+ , .

, 12

,

12

.

4. ( ){ } 2,0 :n n = ] \ .

() .

3

() , ; () ( ),0n ,

( )( ),0 , , 0D n < ( ) ( ) ( ),0 0,0 ,0n n n M = = < , , . n .

5. fD

( ) ( )( ), ln lnf x y x y x= . fD ; fD fD .

( ) ( ){ }( ) ( ) ( ){ }( ){ }( ){ }

2

2

2

2

, : 0 ln 0

, : 0 0, ln 0 0, ln 0

, : 0 0, 1 0 0, 1

, : 0, 1 0, 1 .

fD x y y x x y x

x y y x x y x x y x

x y y x x y x y x x y x

x y x y x x x y x

= > >= > > > < > > > < > + < < < +

\\\\

1

4

6. : 2 2 2

2 2 2

2 2 2 2

2 2 2 2 2

2 2 2 2 2

0 ( 2) ((2 ) 4 9

2 ( ) 0 2 ( ) 02 0 ( 2) 4 4

1.

y z x y zx z y z y z

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

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

() ()() ()() ()() ()() ()

1) 4

4

() x x () (2,0,1) = 2. () : 2 0,x z y z+ = + = 0.() x x . () ( )( )2 2 2 ( ) 0 2 0 0 2 0x y z x y x y x y z x y x y z = + = = + = , 0, 2 0x y x y z = + = . () 2 , ,x y z , ( )0,0,0 . () , . 2 2z x y= + 2

() : 2

2 22

( 2) 12

x y z + + = , 2,1,1. () . ()

7. ( 0)xOy z = 2 2 2

2 2

9, 02 0, 0

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

()() .

9 )

() z

2 22x y+ = ,

( 0xOy z =2 2

2 2222 9, 0 1,3 3

2

x yx y z z+ = = + = = 0

1

0

.

() z( )22 2 22 0 1x y x x y+ = + =

( 0)xOy z = ( )2 +,

. 21 1,x y z= = . xOy . , () :

5

[ ]3 33cos , sin , sin , 0,22 2

x t y t z t t = = = . () :

[ ]1 cos , sin , 1 cos , 0,2x t y t z t t = + = = + 8. ()

2 2

2 2

( , ) 2 ( , ) 1

( , ) ( , , )4

( , , ) ( , , )x y z

f x y x y f x y x y

y xf x y f x y zzx

2

y

f x y z e f x y z x y z+

= + = += =

= =

() ()

() ()

() () +

() ( , ) 2 2,f x y x y c x y c c= + = + = \ ( ) () ( , ) 1 1 , 1f x y x y c x y c c= = + = < ( ) () ( , ) ,yf x y c y c x c

x= = = \ ( )

() 2 2

2 2( , , ) 44

x yf x y z c x y czz+= = + = (

) () ( ) ( , , ) ln , 0x y zf x y z e c x y z c c+ = = + = >() 2 2 2( , , )f x y z x y z c= + = ( )

9. 2 2 2

2 2 2 2

2 2 2

, , 0( , )

,

y x x yx y x yx y

x y

+ + > + += +

6

( ),x y 2 2 4x y 2+ =

e12

.

2 (0,0) , . 10.

: 2 2

12 2 3 3

2 2 2 2( , ) (0,0) ( , ) (0,0) ( , ) (0,0) ( , ) (0,0)

2

2 2 2 2 2 2( , , ) (0,0,0) ( , , ) (0,0,0) ( , , ) (0,0,0)

sin( )lim lim lim lim

lim lim lim

yx y

x y x y x y x y

x y z x y z x y z

x y x y e xx y x y

xyz xy x yzx y z x y z x

+

+ + +

+ + + +

(i) (ii) (iii) (iv)

(v) (vi) (vii)

2 2 2y z+ +

(i) : ( ) 22 0sin, 1F += , 1 .

( ) 2 2, :2

D x y x y = +

7

( ) 2 2 2 22 2 2 2cos 1 2sin 22 2

x y x y 2x y x + ++ = = +

y ,

( )2 2cos 1x y + < , 2 2 2 2x y x y + < + <

= .

2 2

2 2( , ) (0,0)

sin( )lim 1x y

x yx y

+ =+ (ii) : ( ) ( ) ( )3 3 0, cos sinF += 0

( ) ( )2 2

2 22 23 3

2 2 2 2 2 2

2 2 2 2

2

3 2 32

x yx y x yx y x xy yx yx y x y x y

x y x y

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

,

3 3

2 2

x yx y

> = +

+ < + <

,

:

0 > , , (0,1 ) 1 0ln

= > ,

2 2 1ln

x y + < = , 2 2

1x ye + < .

1 , 0 > 2 2

1x ye + < . . : 2 2

1

( , ) (0,0)lim 0x y

x ye +

= .

8

(iv) ( ), yf x y x= . (1 1, 0n n

= nx ),0

( )1

11 1 1 1, lim 1

n nnn n

n

f nn nn

+ + = = = = nx ,

(1 1, 0ne n = ny ),0 , :

( ) ( )1

11 1 1 1 1.

n

nnn n

fe ee

+ = = = ny e

. .

(1 1, 0,na n = nz )0

, 1a >

( ) ( )1

11 1 1 1 1.

n

nnn n

fa aa

+ = = = nz a

(v) (1 1 1, , 0,0,0n n n

= nx )

( ) 32

11 03 3 n

nfn

n

+= = nx .

0 .

: ( ) 2 2 2, , 0 xyzf x y z x y z =

9

( ) 22

11 1

3 3 3nnf

n

+= = nx , ( ) 22

22 2

11 11 11 3nnf

n

+1= = nx .

, . (vii)

(1 1 1, , 0,0,0n n n

= nx ) ( )1 2 3, , 0,0,0n n n

= ny

( )2

0 0 03 nf

n

+= = nx , ( ) 22

55 5 011 11 11n

nf

n

+

= = nx .

, .

( ) ( ) ( ) ( ), , , , , 0,0,0 0t t t = r > ,

( )( ) 2 202 2 2 2 2tf t 2 = + + + +r , . ., . 11. :

(i)

2

2 22

2 2

2 , ( , ) (0,0)( , )

0, ( , ) (0,0)

xx yx e x yf x y x y

x y

+ = + =

(ii) , 0

( , ), 0

x y x yx yf x ya x y

+ += + = , .a\

(i) ( , )f x y ( ) ( ), 0,x y 0 . ( )0,0 ( ) ( )

( , ) (0,0)lim , 0,0 0

x yf x y f = = .

, ( )2 2: 2x k y k + = ( ) ( ) [ ]cos , sin , 0,t k k t k t t = r ( ) ( )0 0,0=r ,

( )( ) 2 21 101 1k ktf t e ek k

= r ,

10

, ( ),f x y

. ( )0,0(ii) ( , )f x y ( ) ( )0 0 0, , ,x y x x x \ . ( )0 0 0, ,x x x \ , 0x y+ = :

( ) ( )0 0

0 0( , ) ( , )lim , ,

x y x xf x y f x x a = = .

, ( ) ( )0 0, , 0,t x t x t t = + + + r 0( ) ( )0 00 ,x x= r

( )( ) ( )( ) 00 00

, 02

, 0t

xx tf t

t x

=+ += +

r

( ),f x y ( )0 0 0, ,x x x \ .

1/2011