www.bakouros.gr

A

f : 2

( ) ln , 0,2

xf x x x

x

.

1. f .

2. ( ) ,f x a a ,

.

3. : 1 ln2 ( ) 2f e f x .

4. : 1

1

( )lim

1( )

x

f x x

x f xx

B

f , , :

2 1( ) 3 2 , 0.xf x x x xx

1. f(x).

2. : lim ( ) lim ( )x x

f x f x

3. f .

4. ( ) 2 ( )g x x f x

f , R

f(2)=2. g : 3( ) ( )( ) ( )g x f f x f x

1. g .

2. 1 3( ) ( )g f x f x x : 3( ) 6f x x 3. f(1)=3, f g.

4. k : ( ) (2 ) 3f k f k k , :

( ) ( ) 2

2

f x k f x kx k x k

, (k,2k).

z : | 5 | 6 | 5 |z i z i , :

1. z.

2. f(x)

, , yo.

www.bakouros.gr

3. , :

2( ) 1 ,g x x x ax x : y=yo

R f :

( ) ( ) ( 1), , .f a f x f a x a

1. f .

2. f (-1, ).

3. 1 , 2 (-1,) :

2 1

( ) ( 1)1

( ) ( )

f a f af f

g >0 :

0,4

164)(,

24)( x

xxxg

xxxg g(1)=-2-, g (1)=-8.

1. .

2. g.

3. : .ln6)(

4)(lim

2 xxxxg

xxg

x

1. [,] h(x),g(x) : ( ) ( ),h x g x :

( ) ( ) , [ , ].a ah x dx g x dx x a

2. f : ( )( ) 1 (0) 0,f xf x e x f :

i. f (x) f(x).

ii. : ( ) ( ) 0.2

xf x x f x x

iii. f,

x=0, x=1, : 1 1

(1).4 2

E f

5 3( ) , .f x x x x x

1. , f. f

.

2. : ( ) (1 ).xf e f x

3. f (0, f(0)).

www.bakouros.gr

4. f 1,

x=3.

f f(1)=1. x : 3

*

1

1( ) ( ) 3 ( 1) 0, , ,

x

g x z f t dt z x z a i az

, :

1. g R g (x).

2. : 1

| | .z zz

3. : 21

Re( ) .2

z

4. f(2)=>0, f(3)=, >, (2,3), ( ) 0.o ox f x

f R , : ( )2 ( ) , , (0) 0.x f xf x e x f

1. : 1

( ) ln2

xe

f x

2. : 00

( )lim

x

x

f x t dt

x

3. : 2014

2012( ) ( ) ( ) .2014

x

x

xh x t f t dt g x

( ) ( ), .h x g x x

4. 20121

( )2015

x

xt f t dt , (0,1).

f(x)=|xz+1|, z =+i , .

1. f , .z I

2. Cf ( ) (0,1), z .

3. ( ) , ,z xf x e x z.

4. 2( ) ln( 1)g x x x 1

( )f x, z .

[1,] > 1 f, g f(1)=g(1)=1

1( ) ( ) ( ) ( ) , [1, ].

a

f x g x f t g t dt x a :

1. ( ) ( )

( ) ( )f x g x

f x g xa x

(1,).

2. 2 3

( ) 11

af x

a

(1,)

3. 1( ( ) ( )) 4.

a

f x g x dx

www.bakouros.gr

: (0, )f , f(1)=1 f(2)=2, :

2 2

1 1( ) ( ) , (0, ).f xt dt f t dt x :

1. (1,2) f()=3/2 .

2. , (1,2)

www.bakouros.gr

: ( ) ln( 1) ( 1)ln , 0.f x x x x x x

1. : 1

ln( 1) ln , 0.x x xx

2. f .

3. : 1

lim ln 1 .x

xx

4. >0 : 11a aa a

R f : 2

0( ) ( ) ( ) (0) 2.f x f x dx f x x f

1. f.

2. f .

3. f,

x=0 x=2.

: 22 1 1( )(x ) ln( x)

f(x)x

1. f ,

0xlimf(x) ,

.

2. f .

3. 1

11

11

2

x

xlim e f(t)dt

2 31

0

(x t )f(x) t e dt

1. f .

2. f .

3. f y=0 .

4. g(x)=f(x)+x, g gC

1gC

f , R, :

2( ) 4 ( ) 3 , .f x f x x x

1. f(0)=0.

www.bakouros.gr

2. :

2

( )limx

f xx

3. .

4. f .

, , ||=||=||=1

|3 ++|2+|+3 +|2+|++3 |2=12. :

1. ++=0

2. |-|+|-|+|-|= 3

3. , , 3 3

.4

E

2( ) 2 ( 2) 2.f x x x

1. f 1-1.

2. f .

3. f f-1 y=x.

4. f f-1.

f (0, ) :

2

1

1( ) 1 ( ) , 0

1

x

f x x f t dt xx

1. f(1)

2. ( ) 3 1f x x

3. f.

f ,

x=2 x=4.

: .,1)( 2 xxxxf

1. f .

2.

2

0 2.

1

1dx

x

3. .)(

)()(

xf

xfxg

4. : .0,,1)2

(2

)1)(1( 222

aaa

aa

f(x)=exx2(x-), >0.

www.bakouros.gr

. 0

www.bakouros.gr

f [2,4], f(2)=1/2, f(4)=1

( ) [ 2,2], [2,4].f x x g(x) : ( )( ) ( ), [2,4].f xg x e x f x x

:

1. xo (2,4) g (xo)=0.

2. ( )( )( ) ( ) 0f xf x e x f x , .

: ( )( ) ( ) ( ) 2 .fg f e f

f R, f(0)=0, ( ) 0,f x :

2

0( ) ( ) ( ) 12 .

x

f x f t dt f x x

1. f .

2. g(x)=f(x)+lnx, x>0, g

.

3. 0

www.bakouros.gr

1. f [-2, 2].

2. 1 4

( 2,2) : ( ) ln4 3

f .

3. : 2

( )

2

11 3.

8

f xe dx

f, g

2

2( ) , ( ) 1 , .

2 4

x xx x e ee ef x g x x x

1. (0,1) : ( ) 0. g

2. 2 21

( ) (2 ) ( ).4

g f f

3. ( ) 4 .g x x

4. k : 2 2( ) 4 ( ) 4g k k g k k

. ln 3 0.

. : 1

( ) 1 (ln 2), 0.f x x xx

1. : 2

1( ) 0, 0.f x x

2. ox : ( ) ( ) 0.o of x f x

1 2 3 1 2 3 1 2 3, , | | | | | | 1. 1z z z z z z Av z z z :

1. 1 2 3

1 1 11

z z z

2. 1 2 2 3 1 3 0z z z z z z

3. 2013 2013 2013

1 2 3

1 1 11

z z z

: 2

2

1( ) , .

1

x

x

ef x x

e

1. .

2. :

2 22013 1 2012 1e e

3. : 2

( ) 1 ( ) .f x f x x

4. , : 1 2

2 2

0 0( ) , 1 ( ) .A f x dx B x f x dx

www.bakouros.gr

f [0,1] f(0)>0.

g [0,1] g(x)>0 [0,1].x

: 0 0

( ) ( ) ( ) , [0,1] ( ) ( ) , [0,1].x x

F x f t g t dt x G x g t dt x

1. F(x)>0 (0,1].

2. : ( ) ( ) ( ), (0,1].f x G x F x x

3. : ( ) (1)

, (0,1].( ) (1)

F x F x

G x G

4. :

2

2

0 0

0 5

0

( ) ( )

lim( )

x x

xx

f t g t dt t dt

g t dt x

f : 1

0( ) ( ) {0,1}.

x

xf t dt f t dt x

:

1. 1

0( ) 0f t dt

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

3. 1

( ) ( ) ( ) ( )x

f x f t dt f x f x (0,1).

. 0x : ln(1 ) .x x

. : ( ) ln( ) , [0, ), 0.xf x e kx x x k

1. :

( ) 0.k

f x xe

2. () Cf, =0, =,

( >0), : 0

( ) xE k x e dx

3. : lim ( ) .E k

g : 2( ) 1, [0,1], , 0 2 3 6g x x x x

[0,1] f. :

1. (0,1) ( ) 0.p g p

2. 1

0( ) ( ) 0.f x g x dx

3. 1 1

2

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

www.bakouros.gr

1 2 3( ) ( ), ( ) ( ), ( ) ( )z f i g z f i g z f i g , f, g

[,]

www.bakouros.gr

3. z,

Cf (0, f(0)) : ln2

ln2013.4

y x

f (0, ) : 1

1

1 ( )( )

x

f xtf x dt

x t .

:

1. f (0, ) .

2. ln 1

( ) , 0.x

f x xx

3. f.

4. ()

f, =1 = 0

www.bakouros.gr

2. 21

, , 0 2 | | 4 Re2

z i z z

: | 1| | |z z

. : 2 4Re( )

0 02 | | 2 | 1|

k k zx xz e dx z e dx

. g ,

g 1-1.

. f [,].

1 2 1 2

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

f t dtx x f x f x

. f : ( ) ( ) 3 2 xf x f x e .

f 0 g 1, 21

( ) | 2 | , :x

g x t t dt

i. f.

ii. : lim ( ) lim ( )x x

f x f x