# Θεματα μαθηματικα κατευθυνσησ γ λυκειου

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• 0

: . . 0

2012

: ,

• 1

: . . 1

1. , :f g \ \ ( ) 5( ) 1 ( ) (1)f g x x x g x= + + + x\ . ) g 1-1.

) 0x \ , ( )0 0f x x= . ) f g ,

g xx.

) ( ) ( )1 2 1 2g x g x x x= = ( ) ( ) ( ) ( ) (1)1 2 1 25 5

1 1 1 2 2 25 5

1 1 2 2

( ) ( )

1 ( ) 1 ( )

1 1 (2)

g x g x f g x f g x

x x g x x x g xx x x x

= = + + + = + + +

+ + = + +

: 5( ) 1x x x = + + 4( ) 5 1 0 ( ) ( ) "1 1"x x x x = + > /

"1 1"

1 2 1 2(2) ( ) ( )x x x x = =

) ( )x

[ ] ( ).( 1) (1) 0( 1) 1

1,1 : ( ) 01,1(1) 3

• 2

: . . 2

2. , :f g \ \ : (1) (2) ... (2006) (1) (2) ... (2006) (1)f f f g g g+ + + = + + + . [ ]0 1, 2006x , : 0 0( ) ( )f x g x= .

( ) ( ) ( ) ( )1 (1) (1) (2) (2) ... (2006) (2006) 0 (2)f g f g f g + + + = : ( ) ( ) ( )h x f x g x= [ ]1,2006 , - [ ]( ) , 1, 2006m h x M x

[ ][ ]

[ ]

( ) (2)

1 1, 2006 (1)

2 1, 2006 (2)2006 (1) (2) ... (2006) 2006

...2006 1, 2006 (2006)

m h M

m h Mm h h h M

m h M

+

+ + +

2006 0 2006 0m M m M 0 ( )h x , ( )h x ,

[ ]0 0 0 01, 2006 : ( ) 0 ( ) ( )x h x f x g x = = 3. f [ ],a

( ) ( ) 0f a f > . ( )0 ,x a ( )0 0f x = . [ ],x a ( ) ( ) 0f x f a .

[ ],x a ( ) ( ) 0f x f a , [ ], : ( ) ( ) 0a f f a <

[ ] ( ). .

1 1

( ) ( ) 0, : ( ) 0

,f f a

a ff a

< =

, [ ] [ ). .

2 2

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

,( ) ( ) 0f ff a f

fff f a

= <

f ( ),a , .

• 3

: . . 3

4. [ ]: ,f a \ : 2( )f > . : 3 33 ( )f x dx

< , 2( )f x x= .

3 3

3 :

32

3x x dx

= : 2( ) ( )g x f x x= [ ]0 0, : ( ) 0x a g x =

2 2( ) ( ) 0 ( ) 0f f g > > >

3 33 3

2 2

3 ( ) ( )3

( ) ( ( ) ) 0

( ) 0

af x dx a f x dx

f x dx x dx f x x dx

g x dx

< < < < .

f f f f f ( [ ],a )

( ) 0( )

( )f x

f x f x

• 4

: . . 4

( ) 0 ( ) ( )(1) ( ) ( ) ( ) ( ) (2)( ) ( )

f x f a ff a f f f af a f

= =

: [ ]( )( ) ,( )

f xg x af x

= ,g :

( )( )

( )2

2 2

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) (3)( ) ( )

f x f x f xf x f x f x f xg x g xf x f x

= =

(2)

( )( )( )

( ) ( )( )( )( )

f ag af a

g a gfgf

= ==

[ ]( ) ( ). . 0 0

,

, , : ( ) 0( ) ( )

Rolleg

g x g xg a g

=

=

( ) ( ) ( )( ) ( )

20

2(3)0 0 0

0 20

( 02

0 0 0 0 0

0

( ( ) ( ), : 0

(

( ( ) ( ) ( ) ( ) 0f x

f x f x f x x a

f x

f x f x f x f x f xx

>

=

= >

6. f \ ( ) ( ) 0 (1)f x f x+ + = , x\ , * \ . ( ) 0f x = .

(1)

(1) ( ) ( ) 0 ( ) ( )

(1) ( ) ( ) 0 ( 2 ) ( )

( 2 ) ( ) (2)

x x

x x

f x f x f x f x

f x f x f x f x

f x f x

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

+ =

( ) ( )2(1) 2 2 0 ( ) ( 2 ) 0( ) ( 2 ) 0 ( 2 ) ( ) (3)

(2), (3) ( 2 ) ( ) ( 2 ) (4)

x xf x f x f x f x

f x f x f x f xf x f x f x

+ + = + = + = =

= = +

( )f x 2T = . 0 (0) (2 )x f f = = ,

• 5

: . . 5

[ ]( ) ( ). .

0, 2

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

Rollef

f ff f

=

=

(4) ( 2 )( 2 ) ( ) ( 2 )( 2 )( 2 ) ( ) ( 2 )

f x x f x f x xf x f x f x

= = + + = = +

f 2T = . , f ( )0, 2 , , 2 ( ).

( ) 0f x = , .

7. f [ ]1, 4 . (1) 1, (2) 2, (3) 3f f f= > < (4) 4f = , ( )0 1, 4x

0( ) 0f x = .

(1) 1(4) 4

ff

= = : ( ) ( )g x f x x= , ( ) ( ) 1g x f x =

(1) (1) 1 0(4) (4) 4 0(2) (2) 2 0(3) (3) 3 0

g fg fg fg f

= == == >= , \ : ( ) ( )f f = .

( )1 i A + , 1 1

11

1 1 1

11

: ( ) (1 ) ( )

( ) (1)( )

x x

xx

x e if x i e if x i

ee f x

f x

+ = + + = += ==

\

( )1 i A + ,

• 7

: . . 7

22 2

2 2 22

: ( ) (1 ) ( ) (2)( )

xx xex e if x i e f x

f x

= + = + ==\

1

1 2

2

1

1 2

2

( ) 1( ) ( )(1), (2)

( ) 1

x

x x

x

f xf x f xe

f x e ee

= =

=

: ( )( ) xf xg xe

=

2

( ) ( ) ( ) ( )( ) ( ) (3)( )

x x

x xf x e f x e f x f xg x g x

e e = =

1

2

11

1 22

2

( )( )( ) ( )

( )( )

x

x

f xg xe g x g x

f xg xe

= =

=

[ ]( )

1 2(3).

1 2 1 2

1 2

,( , ) , : ( ) 0 ( ) ( )

( ) ( )

Rolleg x xg x x g f fg x g x

= =

=\

10. :f \ \ ( ) ( ) 1 (1)f x f x = x\ . (0) 1f = , :

) ( ) ( ) 1f x f x = x\ ) ( ) xf x e= x\ ) fC : 1y x = +

) (1) ( ) ( ) 1 (2)x x

f x f x =

( )( )

(1), (2) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0

( ) ( ) ( ) ( ) 0

( ) ( ) 0( ) ( )

f x f x f x f xf x f x f x f x

f x f x f x f x

f x f x xf x f x c

= =

+ = =

=\

0 (0) ( 0) 1x f f c c= = = , ( ) ( ) 1 (3)f x f x = , ( ) 0, ( ) 0f x f x

• 8

: . . 8

) 1(3) ( )( )

f xf x

=

1

01 1

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

0 : (0) 1

xf x f x f x x f x C ef x

x f C e C

= = == = =

\

( ) xf x e=

) ( )0 0, ( )A x f x . : 00( ) xf x e= , 0

0( )xf x e = .

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

0

0 0

0 00 0

0 0 0 0

( ) 1 1 0( ) ( ) 1 1 0 1 0 1 1

x

x x

f x e xf x x f x e x e e e = = = + = + = + = + =

1y x= + ( ) ( )00, 0,1e = 11. f [ ],a ( ), ( ) ( )f a f = . ( )1 2, ,..., ,a ,

1 2( ) ( ) ... ( ) 0 (1)f f f + + + =

i (1) [ ],a

.

: x + =

• 9

: . . 9

1 1

1 1 1

x

x

x

x x x

= = + =

= = =

f .

f ...

( )

1 1

2 2

3 3

( ) ( ), : ( )

( 2 ) ( ), 2 : ( )

( 3 ) ( 2 )2 , 3 : ( )

...

( ) (1 , : ( )

f a f aa a f

f a f aa a f

f a f aa a f

f f aa f

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

+ + = ( )1 )

+

( )1 2

1 2

( ) ( ) ... ( )

( ) 2 ... ( ) 1

( ) ( ) 0( ) ( ) ... ( ) 0

f f f

f a f f f a f f a

f f af f f

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

+ + + = = =

• 10

: . . 10

12. f [ ],a , ( ) 0f = . ( ) 0f > , [ ],a f

.

f f f f f ( ) 0( ) ( ) ( )( ) 0 lim 0 lim 0

f

x x

f x f f xfx x

=

> > >

( ) 0f xx > .

( )0 : , x > ( ) 0f xx > .

( ),x ( ) 0f x < 0x < , . ( )( ) 0 ,f x x < f , [ ]( ) ,f x x 0 .

13. 1x

x xI dxe

= + . : i)

1

x

x

xe xI dxe

= + ii) I =

i) : ( )x y x a y = = + x y dx dy dy dx= = = x y yx y y

= = == = =

( ) ( )( ) ( ) 11 1 1x y yy yx x y yI dx dy dy

e ee

= = = =+ + +

1 1 1

y x

y y x

y

y y ye y xe xdy dy dxe e e

e

= = =+ + +

ii) 1 21

1

xx

xx

x

x xI dxx x xe xe I dx

exe xI dxe

+

= ++ = += +

• 11

: . . 11

( ) ( ) ( )( ) ( )( ) ( ) [ ]

(1 )2 21

2 2

2 ( ) ( )

2 ( 1) ( 1)

2 ( ) 2 2

x

x

x x eI dx I x xdxe

I x x dx I x x x dx

I xdx

I x

I I I

+ = =+ = =

= + = + + = + + = =

14. : /2

0

( )( ) ( ) 4

x

x xxI dx

x x

= =+ .

: 2

x y dx dy dy dx= = =

0 02 2

02 2 2

x y y

x y y

= = = = = =

2

2

x y y

x y x

= = = =

, ( )( ) ( ) ( )0

/2

y

y y

yI dy

y y

= +( )

( ) ( )/2

0

y

y y

yI dy

y y

= + ( )

( ) ( )( )

( ) ( )

( )( ) ( )

/2

0 ( ) /2

0/2

0

( )2

x

x x x x

x xx

x x

xI

x x x xI dx

x xx I dx

x x

+= + + = += +

/2

02 2 0

2 4I dx I I

= = =

• 12

: . . 12

15. f \ , 2( )f x x > x\