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(1)
: f :
2 2( ) 2 ( ) 2 0f x f x x + = , fx D , :
:
( )22 22
( ) 2 ( ) 2 0 ( ) 1 1
( ) 1 1
f x f x x f x x
f x x
+ = = =
2
, ( ) ( ) 1g x f x= ( , 1] [1, + ) 1 1x x= = .
,
(1 .
( , 1 )
)
, )+
(1( , 1 , )+ , :
( ) ( ) 1g x f x=
2( ) 1 1f x x = , 2( ) 1 1f x x= +
(1( , 1 ) , )+ , :
( ) ( ) 1g x f x=
2( ) 1 1f x x = , 2( ) 1 1f x x=
,
, :
. , ..., 2008 1/11
) 2( ) 1 1, ( , 1] [1, )f x x x= + + ,
) 2( ) 1 1, ( , 1] [1, )f x x x= + ,
. , ..., 2008 2/11
) 2
2
1 1, ( ,( )
1 1, (1,
x xf x
x x
+ = +
1)
)
) 2
2
1 1, ( ,( )
1 1, (1,
x xf x
x x
=+ +
1)
)
. , ..., 2008 3/11
(2).
f : 3( ) 3 ( ) , f x f x x x = \ .
) f 1-1
.
) f
;
R
:
) 1 2,x x R , 1( ) ( )2f x f x= . 1 1 1 2 2
3 3( ) 3 ( ) ( ) 3 ( )2x f x f x x f x f x= = . f 1-1. 1x x= 2 f ( )y f x= x\ , x :
3 3x y y= f , :
11 3( ) 3 , ff x x x x D =
1f , R 1f 1-1,
3( ) 3 ,g x x x x= R 1-1.
) f . ,
f ,
,
R
R1f ,
f, .
. , ..., 2008 4/11
1f
, R 1f
f. 1 3( ) 3 ,f x x x x = 1-1, o
: R
( ), 1 , 1, 1 1, + . ,
3( ) 3g x x= x . f R .
R
f
, ,
( ) (0 , 2 2,x +)[ ]0 2, 2x .
) f:
0x R . 0, x x xR , : 3 ( ) 3 ( )f x f x = x 0 3 0 0( ) 3 ( )f x f x x = ,
. , ..., 2008 5/11
, , :
( ) ( ) ( )2 20 0 0 0 (1)( ) ( ) ( ) ( ) ( ) ( ) 3f x f x f x f x f x f x x x + + = :
( )2 20 0( ) ( ) ( ) 3h z z f x z f x= + + , ( )2 03 ( ) 4f x = :
( )2 0( ) 44
3min ( ) 4
f xm h z
= = = , :
( )2 20 0( ) ( ) ( ) 3 , h z z f x z f x m z= + + R :
( )2 20 0( ( )) ( ) ( ) ( ) ( ) 3 , h f x f x f x f x f x m x= + + R ,
A , 0 0( ) 2 ( ) 2f x f x< > ( )1 , , , (1), :
0 02 2x x< >0min ( )m h z >=
( )2 20 0( ) ( ) ( ) ( ) 3 0, (2)f x f x f x f x m x+ + > R :
( )0 0 02 20 01 1( ) ( ) (3)
( ) ( ) ( ) ( ) 3f x f x x x x x
mf x f x f x f x = + +
0
0lim - 0x x x x = . 0 0lim ( ) ( )x x f x f x = , f
0x .
f ( ) ( )0 , 2 2,x + . ( )1
3( ) 3 , g x x x x= R( ], 1 [ )1,+ [ ]1, 1 . :
0 0 0 0
0 0
( ) 2 ( ) 2 ( ( )) ( 2) ( ( )) (2)
2 2f x f x g f x g g f x g
x x
< > < > < >
. , ..., 2008 6/11
A , 02 ( )f x 2 2 02 x ,
, f
min ( ) 0m h z= 0x .
f
, [ 2, 2] f
[ 2, 2] .
) f:
A , , (1) (2), : 0 02 x x< > 2
( )0 2 20 0 0( ) ( ) 1
( ) ( ) ( ) ( ) 3f x f x
x x f x f x f x f x = + +
f 0x , :
( ) ( )0
0
00
0
22 200 0
( ) ( )( ) lim
1 1lim3 ( ) 1( ) ( ) ( ) ( ) 3
x x
x x
f x f xf xx x
f xf x f x f x f x
= = = + +
f ( ) ( ), 2 2,x + . , ,
f
02 x 20x .
f
, [ 2, 2] f
[ 2, 2] .
. , ..., 2008 7/11
(3).
f :
f x f x x+ = 3( ) 3 ( ) , x R . f:
) 1-1 .
)
, R R ( ], 0 [ )0, + .
:
) 1 2,x x R , 1( ) ( )2f x f x= . 1 1 1 2 2
3 3( ) 3 ( ) ( ) 3 ( )2x f x f x x f x f x= + = + . f 1-1. 1x x= 2 f ( )y f x= x\ , x :
3 3x y y= + f , :
11 3( ) 3 , ff x x x x D = +
f
.
0y R 0x R , 0( ) 0f x y= . . 0y R 0 03 3 0x y y= +
. , ..., 2008 8/11
0x 0( ) 0f x y= . , :
30 0 0 0 0 0
3
0
0
0 0
3 3
0
3
3(,
( ) 3 ( ) ( ) 3 ( )( ( )
1 1( )) ,
( )) g f g
f
f x f x x f x f x y yx y g x
x y g
++
+ = + ==
=
=x x
, 1 3( ) 3 , f x x x x = + R .
1) f:
0x R . 0, x x xR , : 3( ) 3 ( )f x f x+ x= 0 3 0 0( ) 3 ( )f x f x x+ = ,
, , :
( ) ( ) ( )2 20 0 0 0( ) ( ) ( ) ( ) ( ) ( ) 3 (1)f x f x f x f x f x f x x x + + + = :
( )2 20 0( ) ( ) ( ) 3h z z f x z f x= + + + , :
( )2 0( ) 4min ( ) 3 04f xm h z += = > , :
( )2 20 0( ) ( ) ( ) 3 0, h z z f x z f x m z= + + + > R :
( )2 20 0( ( )) ( ) ( ) ( ) ( ) 3 0, h f x f x f x f x f x m x= + + + > R , :
( )0 0 02 20 01 1( ) ( ) (2)
( ) ( ) ( ) ( ) 3f x f x x x x x
mf x f x f x f x = + + +
. , ..., 2008 9/11
0
0lim - 0x x x x = . 0 0lim ( ) ( )x x f x f x = , f . 0 Rx f . R
2) f:
0x R . 0, x x xR , , , :
( )0 2 20 0 0( ) ( ) 1
( ) ( ) ( ) ( ) 3f x f x
x x f x f x f x f x = + + +
f 0x , :
( ) ( )0
0
00
0
22 200 0
( ) ( )( ) lim
1 1lim3 ( ) 1( ) ( ) ( ) ( ) 3
x x
x x
f x f xf xx x
f xf x f x f x f x
= = = ++ + +
f R
( )21( ) 0 (1)
3 ( ) 1f x
f x = >+
, f , R (0) 0f = , :
( )( ) 0, , 0f x < , (0) 0f = & ( ) (2)( ) 0, 0,f x > + f , (1), R f R :
( )322 ( ) (3)
9 ( ) 1( ) f x
f xf x =
+
. , ..., 2008 10/11
, (2), f ( , 0 , )0, + (0,0)
13
y x= .
. , ..., 2008 11/11