E

1

z : ( ) ( )4Re 2Re , 1z zz

+ =

. z . . ( )Re 0z , :

i. 4w zz

= + 4 4w .

ii. 3 4c z i= + + .. , c .

. 1 2,z z 3z (1) ,

1 2 2 3 3 1 1 2 32z z z z z z z z z+ + = + +

2

3( ) 4( ),zz z z i z z z C+ = + (1) : . z

. z 1z

. , R , 1z 21 0

4z z + + =

. 0z (1), 2012

20120 4 3 5 , 55

z i w iw i

+=

,

w (0,5) 2 1 = .

30

+ 10

3

z C , 1z = 1z a+ = , a R . : . 0 2a

. 2 2( )2

aRe z =

. 2 21 3z z a + =

. 2133 14

a z z a +

4

,z w 2 2| | 1,| | 3z zw w zw+ = + = . . | | 2z w+ = . . z w , . . z w . . ,z w .

5

2 2(cos ) (5 4sin ) 0, [0, ]z t z t t + = . : . 1 2,z z

. 1 2| |z z

. 1 2| |z z+

6

( 1) , [0,1]z t t i t= + . . z . | |z

2 2( 2) ( 1) ,w k k i k= + + , : . w . | |w . | |z w . | |w | |z w [0, 4]k

7

. 2 1 0w w+ + = . 1 2,z z

2 21 1 2 2 0z z z z+ + =

i. : 1 2z z=

ii. : 1 2 1 2z z z z+ = =

iii. *N 1 2 0z z + , 1 2

1 2

z zuz z

=

+ .

8

1 2 3, ,z z z , ,A B ,

: 1 2 32 3z z z+ = 1 3 21, 2z z z= = =

. 1 2Re( ) 0z z =

. i. 2 2 2

1 2 1 2z z z z = +

ii. OAB .. 2 3Re( )z z 1 3Re( )z z. i. , ,A B .

ii. A B

9

1 2,z a bi z c di= + = + , , ,a b c d 1 2 1z z= = .

2 1 22 2 0x z z x + = 1 2,x x . :

. 1 2,x x .

. 1 2 2x x= =

. 2 2

1 2 1 24 8x x z z + =

. 1 22 1

x xwx x

= + .

10

( ) ( )6 8 , 0z t t i t = + + + .. 6 8z i . . z . . z . 0t , z 1 3

. 0t = R , 11

w zz

= + ++

.

11

2( )(1 )x y iz

x yi+ +

=+

*x, y R .

. ( )2 2

2 2

22 x y xyRe zx y+ +

= +

( )2 2

2 22x yIm zx y

=+

*x, y .

. z . . z . . z .

12

z 2| 1| | 9 20 |z z z = + | 4 | , 0z = > . . :

i. 2 2| | 16

4zz z + + =

ii. 2 2 2 2(1 ) | | (5 1)( ) 25 1z z z + + = iii. | | 2z

. z . . z .

13

,z w C 2 22 2(3 4) 5 0, (4 3 ) 5 0z i iz w i iw+ + = + + = .

. 2 2(4 3 ) 5 ( ) 0z w i i z w+ + + + = .

. q 1 1qz

= + .

14

,z w C 1zw = .

, , ,a b c d R 2 2 2 2 0a b c d+ + > ( ) ( ) ( 1) ( 1) 0a z z ib z z c zz d zz+ + + + + = , : . z . z , w . . z , w

.

15

, *a C 1 2,z z 2 2 0z az + + = . :

. | | | | 1a = = 1| | 2z 2| | 2z .

. 1 2| | | |z z= , a

.

. 2a R , 1

2

zz

, 1 2| | | |z z=

16

*22( 2 ),z z Cz

= (1)

. 1z 2z (1)

. v , 1 2 0v vz z+ =

. x y ,

2011 162 21 2

1 1 1( )i ix yi z z

+ = + ++

. z , 2 41 2z z z z =

. 0z .

. 7z i+ .

17

( ) ( )z k t k t i = + t R 1k > . . z .

. w ( 1)y x k= , k 5 2| | 12min

z w =

. k () z | |z z . k () | 3 4 |w i min + . u ( 1 ) ( 1 )u m t m t i = + + + , m u . . ,k m () () , | |z u .

18

z, w 2 24 | | 2 1,| | 2 3z zw w zw = = . . | 2 | 2z w = . . z w , . . | 6 |z w+ . . z w .

19

z ( )| 2 | | 1|

if zz z

=

. z ( )f z . . | ( ) | 1f z . ( )f z i= , :

i. N | 1| ( ) 1z Re z + =ii. ( )Re z .iii. z , z

.i.

20

. ( 3 )( 4 2 )z i z i+ + +. 2 (1 3 ) 14 2 0z i z i + = (1). 1 2,z z (1) 1Re( ) 0z > , ,A B 1 2,z z 3 3z i= + . AB . M( )z z

2 2 2( ) 2( ) 2( ) 30MA MB M+ = +

21

1 2,z z 12

z zz

= 1 2, ( ) 0z Im zz

+ = .

. z . . *N 2 21 2z z

= .

. 95 94 ... 1 0z z z+ + + + = . 2z 2 1y x= + , 1z , . . OAB , , ,O A B 1 20, ,z z . . OAB () .

22

{2 }z C i 3 8( )

2z if zz i+

=

. 2z i , 2( ) 2 4f z z iz= +

. (1 )f i+

. 2( ) 2 4f z z iz= +

. ( ) 2 6f z iz z= +

. 1z = , ( )f z 7 =

23

z ( ) ( )2005 2008 1 1z z = .. z

. 2z z= . (1)

. z ( )Im 0z > , (1) zwz

+

=

R .

i. w

ii. , w

24

1 2,z z 1( ) 0Im z > 1 2 2 1| | | | 40(1)z z z z+ = 1 2 25(2)z z = .

. 1z 2z .

. 1w z i= , .

. *N 1 2z z z = . 1 2,z z z

.

25

2 0, ,z z R + + = 12zi

= 2z

. , R 2z

. v R , 1 2 16v vz z i =

. z

2 2

1 2 16z z z z + = (1)

. z (1) , 4 4z i

26

, 0z w 0zw zw+ =

. 2010z

w

.

. ,z w .

. z w z w = + .

. 2z w iw z+ =

i. zw

( )1,0 .

ii. 2012z

w

.

27

z : 2 2 2 21 2 | | | 1| 2 | 1|z z z+ = + + +. z . . 2 2( 1) ( 1) 0z z zz+ + + =

. 2 1 0z z+ + = . z . . 2012 2014 2013A z z= + +

28

z 2z i 2 4( )

2zf zz i+

=

(1)

. Im( (1 ))f i+ . z , ( )f z R

. ( ) 2f z z i= + . z , ( 5 ) ( ) 10f z i f z i + + = (2) . z (2) ,

. 1z 2z (2) , 1 28 10z z

29

1 2z , z 2z z+9=0 , R 1 2z , z R .

. .

. ( )17 171 2z z R+ . . 1 2z , z .

. 1 22 1

z z 2z z

+ = .

. =0 ( )1Im 0z > z 1 2z z 4 z z = + .

30

. , , : 3 23 4 2 0z z z + =

. , : 3 24 3 2z z z+ = + 10 32 32 0z z + =

31

f , [ ],a b , 0a > ( ),a b . : ( ) ( )1 2z a if a z b if b= + = +. 1 2 1 2z z z z+ = , ( )1 ,x a b , ( )1 0f x =. A B , A B , : 1 2 1 2 100Az z Bz z+ = . :

i. 1 2z z

ii. ( ) ( )f a f ba b

=

iii. ( ),ox a b , ( )( )o

oo

f xf x

x =

iv. fC

32

f : [ , ] R ( ) 0f > > , ( )( )

ifzif

+

=

. :

. ( )f x x= ( , ) . 0 ( , )x 0( ) 1f x z .

. i. f z , ii. .

.

Bolzano f [ 3, 3]

. ( ) 23 ( 2 1)3

w f z i= + + , w

34

z x yi= + , 22012 ( ) 1 2012 2012 3 0

21Im z i

z+ + =

:

. ( , )M x y z . . z ; . z , ( ) 0Im z < .

35

f R , ( ) 0f x x R 1

( ) 0z

f x dx = f(1) = 1 .. ( ) 0f x > . z .

. ( )( )

3

2

3lim

3xz z x xz z x x+ +

+

. f 'x x 0x = 1x = 2z z+ ,

20

( ) 3 6 6x

f t dt x x= + (0,1)

36

[ ], f ( )2z if = + ( )2w if = 0 ( ) ( ) 0f f . w z w z+ <

( ) ( ) ( )f f f < < . . ( )0 ,x ( )0 0f x =. ( )1 ,x ( ) ( )1f x f =

37

. 1 2,z z 1 2 1 2z z z z+ .

1 2w z z= .

. ( )1 1f xz i= + ( )( )2 1z f x i= + +

() f R ( )0 0f = ( )0 0f . 2 3< <

. g ( ) 1 2( )g x Im z z= Rolle [ ], ( )

( )( )( )

11

f

f

fefe

+=

+, ( ) 1f

38

f : R R (0, 2)A .

( ) ( )z f x f x i= + 2( ) ( )w f x f x i= 2( 1)xz e= +

. f . . z x R . . ( ) Re( )g x z w= .

39

:f R R 12

z C

2 2( ) 2 ( )f x x xf x+ =

x 0

( ) | 2 |,|

m2 1|

lix

f x zl lx z

= =

.

. : i. | 2 | | 2 1|z z = ii. z .

. 20( )lim

x

f xx x

.

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

. 3(| 3 4 | 5) 10z i x x+ + = + [ ]1,2

40

( ) ( )1 3 3 3z x x i = + + + , [ )0,2x .. M z ( )C .

. x z .

z .

. 1 2,x x x z

1 2,M M z . :f R R

1 2,M M . f

'x x ( )C .

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1

. f [0,1] 0 (0,1)x 00 0

1 1( )1

f xx x

=

.

. f R 1 ( ) xx f x e+ x R . :

i. f 0 0x =

ii. f R , 0 ( 1,1)x , 0 0( )

2004f x x= .

iii. lim ( )x

f x+

2

f : 55 3( ) 2 2f x x z x z= + x R *z C .

. f . . f .

. ( ) 0f x = (0, )z

. 5

30

( ) 2lim 1x

f x zx

+= , z .

3

R f :g R R 2( ) ( ) ( ) 1xg x f x e g x= + x R | (0) | 1f < .

. (lim )x

f x

. (lim )x

g x

30

http://www.study4exams.gr/

. (0) 1g

. ( 1)( )[ ( ) ] [ ( 1) ]x xx e x g x e x f x x + + = + [0,1) .

4

:f R R ( ) 112

f = ( ) ( ) ( )3 3f xy f x f f y fy x

= +

*,x y R . :

i. ( ) 132

f =

ii. ( )3 ,f f x x Rx

=

iii. ( ) ( ) ( )2f xy f x f y= ( )2 14

f x = x R

. f .

5

f : R R g : R R x R ( ( )) 2 ( )f f x g x x= . g R . . ( ) ( ) ( )h x f x g x= . 0x R 0 0( )f x x=

i. fC gC .

ii. 0 0 0( ( 2)) 2 ( 2) 2f f x x x x f x x+ + + = + +

iii. 0 0( (ln 1)) ln 1f f x x x x+ + + +

( ) 0f x = . . (1)f . f .

. 2( 2) ( ) (5 6)f x f x f x + = . ( ) 0f x < 1x > , f (0, )+ .

7

( ) 2 ln( 2 1)f x x= + . . . f .

. f 1f

. 1( ) 2f x = . fC y x=

8

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

9

( ) (1 ) (1 )x xf x ln e ln e= + . fD

. ( )f x. f

. 1f . 0m < ( )f m m= . ( ) ( ) , 0g x f x x x= < , ( )g x . ( ) ( 1) 1f x f x < + . ( ) ( )h x ln x= , c R ( ) ( )f c h c=

. : 3 2

2

( 1) 6lim( 3) 2x

f x xAf x x + +

=

, ( )1( ) ( )lim f x f xxB e e

=

10

f ( ) lnf x x x= + . . f . . ( ) 1f x = . f ( ) 0f x = . . f f .

. 1( )f x x =

. 1( ) 1f x x >

11

f 2 ( ) ( )f x f x x =

. 2 ( ) ( )f x x f x

. ( )f x x

. 0

( )lim ( )xf x

f x

. 0

( )lim x

f xx

12

23( ) 3 30 95 (3 5), ,

4 4g x x x x x R R = + + + 4 5( ) ,

3xh x x R=

. f goh= x R 2( ) ( )( ) 5 10f x goh x x x x= = + +

. R lim ( )x

f x+

. R lim ( )x

f x+

42

( ) 2lim14x

f x xx x

++

. R 3

lim( )x

xf x x

+ +

13

:f R R ( )( ) ( ) 1f x f y f x y+ = + + ,x y R . : . f .

. ( ) ( ) 12 ( ) 1f x f x f x= + +. f . . ( 1) 0f =

14

, < :f R R ,

: ( ) 2f = , ( ) 2f = ( ) 2012f x < x R .. ( ) ( )2x f x f = + ( ]0, + . f [ ], = :

i. ( , ) ( )f = + . ii. fC f 2y x= ( ),ox .

. 2( ) 4

1xxf x xlim

x

+ +. :h R R ( ) ( ) 2004f x h x x = , x R .

( ) 0f x = 1 2, . ( ) 0h x = ( )1 2,

15

1 1 , :f g R R 1( ) ( )( ) 8f x fog x = 13( )( ) 2( )( ) 10 7fog x fog x x+ = x R

. ( ), ( )f x g x

. ( ) :h x R R 2( ( ( ))) 4 2xh g f x e x= x R i. ( )h xii. ( )h x .

iii. 22

3 2 6xxe e x x

ee >

iv. : ( )lim

x

h xx+

, 2( 1)lim

1xxh xe+

++

, 1 3 2

2

( ) 2 3lim5x

h e x xx x

+

+ ++

. N ( )h x lnx= .

16

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

. ( ) ( )1f x fx

=

17

:f R R ( ) 1f x ( ( ) )( ( ) 3 )f x k f y k k + = ,x y R k R . . k . . k f .

18

( ) : (0, )f x R + ( ) ( ) ( )f x f y f x y= + 0ox = (1)f e= .

. 1(0) 1, ( 1)f f e= =. N R . . :

i. ;ii. lim ( )

xf x

lim ( )

xf x

+

. (0,1)ox 1 1 13 ( ) (2 ) (3 ) (4 )of x f f f = + +

. : 10

lim ( )x

f x+

, 1lim ( )

xf x

+ ,

0

( 1 )lim( )x

f xf x +

. 1 1 1( ) ( ) ( )f ab f a f b = + , 0a b >

. 1 2

1

( )lim( )x

f xf x

+

. 1 10 : ( )o o ox f x x > =

. 1

1

( )( )lim( )x

f f xf x

19

( ) 3 7 5f x x x= + .. :

i. f 1 1

ii. ( ) 0f x = ( )0,1

. ( )( )

2

3, 1

13 , 1

f xxg x x

a a x

=

+ =

, *a R , ( )g x 1ox =

. i. ( )limx

f x+

( )limx

f x

ii. ox R , ( ) 7of x =iii. ( ), 1k k + , ox R , k

. :

i. ( )

4limxf x x

x

+

ii. , ( ) ( )3 5 2 6f f =

20

f [ ]0,1 2 2(0) (1) 13 6 (0) 4 (1)f f f f+ + = + .

. N : i. f .

ii. 1x 2x (0,1) :

. f 3y x= 1x .

. 212 ( ) 3 (1/ ) 4 (1/ ) 5 (1/ 2)f x f e f f= + +

. 1( ( 4) 1) 3f f lnx + > . ( ) ( )z f x if x= + [0,1]x

i. z .ii. | 5 |z .

21

g 0y > 2( ) lim 1x yyxg x

e x+=

+ +

22

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

. 2( ) ( ) (1) (2)g x f x f f= (1, 2) . . f .

23

. ( ) ( )1

lim limo

ox x hf x l f x h l

= =

. ( ): 0,f R+ ( ) ( ) ( )f xy f x f y= + , ,x y R . :

i. f 1ox = , f ( )0,+ii. f ox a= , ( ) ( )0,1 1,a + , f ( )0,+

iii. f 1ox = ( )

1lim 1

1xf xx

=

, ( ) ( ) 1lim

x a

f x f ax a a

=

,

( ) ( )0,1 1,a +

24

:f R R : ( ) ( ) ( )f x y f x f y + = + , ,x y R ,R ().

. ( )0f =. ( ) ( ) ( )f x y f x f y = + ,x y R .. ( ) ( ) ( )1f x f x = x R , *N .. ( )f x = R , f 1 1

( ) ( ) ( )1 1 1f x y f x f y + = + ( ),x y f R. 0x > ( )f x > , f R

( ) ( )1 2 12 ( ) (3 1)f f x f f x < + .. f 0ox = lim ( )x f x

25

f : R R 3 ( ) 5 ( ) 0f x f x x+ + = x R . f . f

. f R 1f . f R. f R . ( 19) 1f x x = +

. 1

0

( )limx

f xx

26

: Rf R , x R : 2 2( ) 1x f x x< < + .

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

i. 1 1( ) 1( ) x

g xf x e

= + , x R

ii. ( ) ( )x xe f x e f x+ = ( )0,2 .

. 20

1lim lnx

x f xx+

+ .

27

: (0,1)f R 0

( ) 5lim 3x

f xx

=

22 ( 1) ( 1) ( ) 1x x f x x (0,1)x . . ( ) ( ) ln 3,g x f x x= (0,1)x .

. ( ) 3 ( ) f xh x e =

, ( )0,1ox . ( )3 f xxe e+ = ( )0,1 , R

28

:f R R ( )f R R= R,x y 1( ) ( )f x f y x y

,

( )0,1 . :. f ,

. 1 1( ) ( )f x f y x y ,x y R ,

. 1( )f x R ,

. 1( )f x x = R .

29

, :f g R R ( ) ( ) ( ) ( )2 2 1 2 2f x g x xf x g x+ + = + , x R .

. ( )( ) ( )( )2 2 21f x x g x x + = , x R .. f g 0ox =

. ( )

2limxf xx+

. g R , ( ) 2g x x= R

30

,f g ( ) ( )2f x ln x x= ( ) ( )1g x ln x= .. h ,f g

. 1h . ( ) ( ) ( )( ) ( )1 11 2h x h x ln h ln eh e + = +

http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775

1

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

( ) ( )1

2

0

1 tf x x dtf xt

= + , x R . ( ) 0f x > x R. f . f x

. ( ) ( )x

x

tg x dtf t

= x R .

. N ( ) ( )f b f a b a ,a b R

2

11

2

( )x

tf x e dt= .. . . f . . fC (2, (2))A f

. ( ) 2f x ex e 1x > . E fC , 'x x 2x =

4x = , 2E e

60 A

3

f :[0,1] R

1

( ) ( ) ( )x

f x xf x f t dt = + [0,1]x

. f [0,1]

. 0 (0,1)x 1

00

( ) (1) ( )f x f f t dt = +

. ( ) ( )x

f x x f t dt c= + c R (0) (1)f f=

2

(1)( )1

ff

=+

1

. f (0,1) ,

4

2 201( ) ln( 1)

1x

f x x x dtt

= + + +

. f .

. 34

20

10 ln 21

dtt

< 0,2

20

:f R R ( )( )

0

1 1f x

te dt x+ = x R .

. f . f . f x R . fC , 'x x

1x = 1x e= +

. ( ) ( ) ( ) 11 xx f x f x < < 1x >

21

( ) 3xf x e x= ln

1

( 3 )( )3

x t t

te e tg x dt

e t+

=

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

0,3e

( )ln 3,2

. g. g. g , ga b D a b

. ( )1

0

1f x dx =

. ( ) ( )2

0

1x

f t dt xf x= .

24

f : (0, ) R+ ( ) ( ) 1x

e

f t dt f xx

= 0x > .

: . f. ( ) fC

. fC 'x x ( )

.

2

1

( )lim( )

x

xx

f x dtf t+ .

25

f ,g R

01

( )

( )( ) 0

x

x

f t dt

f t dtg t dt >

{0,1}x R g(x) g(2 x) 2+ = g(x) 0 x R .

1

0( ) ( )

x

xf t dt f t dt> .

. 1

0( ) 0f t dt = .

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

. 1

( ) ( ) ( ) ( )x

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

. 1

( ) ( )x

f t dt xf x= ( )0,1. 2 ( ) ( )f x xf x= ( )0,1 f .. ( )g x x x

0x = 2x = .

26

f R 2

2

1

( ) 1x f tx dt x

x x { }0x R

. 3

( ) ( )x

xx f t dt = .

. ( )f 1 1= ( )f -1 1= . ( )x 0x = (0) 0f = . f ,

.

27

:f R R 32

3 ( ) 6 63

x

x

x tf x f dt x x

= + x R .

. f . . f . . f 1 1 . .

f, 1fC , 'x x 22 , 2x xe

= = .

28

f (0, )+ 2 11 1 1( ) 22 2

x xf x f dtx t t

= + 0x > .

. f (0, )+ ( )f x ( )f x .

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

. ( )E k fC , 'x x

1,x x k= = (0,1)k . . N

0lim ( )k

E k+

.

29

f : (0, +) R x > 0 31

1 2 ( )( )x f tf x dt

t t =

. 2ln( ) , 0xf x xx

= >

. f

. ( )E fC , 'x x 1xe

=

x = 0 > , 0

lim ( )E

+

lim ( )E

+

. , R 3 2

2

lnlim 0x

x x xx

+

+ =

30

g : R R f : R R .

, x R ( ) 0f x > ( ) 0g x > . ( )

0

( )( )

xg x tF x f dtg x

=

.

. x R 0

( ) ( ) ( )x

F x g x f u du= . F , ( ) xf x e= ( ) xg x e= . x R ( )F x x , (0) (0) 1g f = . (1) (2) (2) (1)F g F g<

31

: 0, 2f R 2( )f x x = ( )0, 2x

:[0, 2 ]g ( )( )( ) , 0, 2f xg x xx = [0, 2 ) . ( ) ( ) 0f f = = . (0)g

. : 2

( )2

xf xx

( )0, 2x

. : 0

1( )6

xf x dx

=

32

:f R R ( ) 0f x > x R

(0) 1f = .

2 ( ) , 0( )

2 , 0

x

x

f t dt xg x t

ln x

= =

. N ( )( ) (2 )2

g xf x f xln

0x > ( )(2 ) ( )2

g xf x f xln

0x 0 > ( )0

( )x f tf dtx

= . g .

. ( )0,1 1

( ) 2 ( )f t dt f

=

39

( ) ( )70

x

f x t t dt = 0, 2x

. 02

f =

. f , ,

. 1n > , ( )11 ffn n

41

:[ , ]f R ( ) 2 ( )f f = : 2( ) 2 ( ) 4 ( ) 4f x f x f x = + [ , ]x .

: . f . . ( ) 0f > .

. ( ) 2f x dx ln

, : i. f ii. f ,

.

42

f [ ]0,1 [ ]0,1 .

. ( )1

01f t dt

44

:f R R ( ) ( )204

1

x

f x dtf t

=+ , x R .

. f R . f ,

. ( ) ( )3 3 12f x f x x+ =. ( )f x x=

. ( )3

0

f x dx 45

[ ]: ,f a b R 0 a b< < ,

( )1z a if a= + ( )2z b if b= + 12

zw Rz

= .

. 1 2 1 2z iz z iz+ = . Rolle

( ) ( )f xg xx

= [ ],a b. f

. ( )

( )( )lim 1

x

x aa

f x a tdt

x a x a t+

= + , ( ) 1f x = ( ),a b

46

: (0, )f R+ ( ) 0f x 0x > .

( ) 11,1 , , 22

( )

2

1( )1 ( )

f xg x dt

f t=

: . ( )g x . ( )g x . 0x > ( ( ))f f x x= , :

i. '(1) 1f =

ii. (0,1)ox 2'( )

1o og x

x=

iii. (0,1) 3'( )2

f =

47

f : (0) 1f = ( ) ( )f x f x > [0, )x +

. [0, )x + ( ) xf x e>

. [0, )x + 0

( ) 2 3x

x xf t dt e+ +

(0,1) (1, ) +

. 0

( ) 2x

xf t dt e x+ = + (0,1)

48

f :[0, ) R+ , (0) 1f =

1

0

( ) ( ) (1) (1)x

f x f x f f edxe e

+ = .

. (0) 0f = . (0, )x + ( ) 1f x >

. (1) 2f = , 1

0

31 ( )2

f x dx<

iii. ( )2 2

( ) 1

1 1

ln ( )f xe dx f x dx >

49

: (0, )f R+ :

(1) 1f = 2 ( ( ) 1) ln 1x f x x = 0x >

. N ln( ) ( ) , 0xg x f x x

x= + > (0, )+ .

. N f .

. 1

11 xxx e + 1x . fC , 'x x

1,x x e= = .

50

( ) ln( 1) ln , 0xf x e x x= > . 0x > ( ) 0f x >. f .

. 2

1

lim ( )x

xx

f t dt+

++

= +

.

2

1

lim ( ) 0x

x

x

xf t dt+

=

51

f [1, )+ , ( ) 0f x > 1x

2

1

( ) ( ) , 1x

G x t f t dt x= 1

( ) ( ) , 1x

H x tf t dt x=

. ( )( )( )

G xF xH x

= (1, )+

. ( ) ( ) ( ), 1P x xH x G x x= . : i. 1x ( ) 0P x ii. P [1, )+

.

2

121

( ) lnlim

( )( 1)

x

x

H x tdt

G x x+

52

:f R R ( ) ( )1 2 1 2f x f x x + = + = , x R (1) 0f =

. ( ) ( )1 1 2 ,f x f x x x R+ + =

. ( ) ( )1

2

1

x

x

g x f t dt x+

=

. ( )3

1

I f x dx

=

53

( ) 22

x

f x u u du= .

. f. f ,

. ( ) ( )23 2

x ttg x e u u du dt

=

. ( )1 2, 1,3 ( ) ( ) ( )1 23

1 21

te f e e f t dt + =

. : ( )( )

3

3

33

xt f te dt x

f

( )2,x +

54

( ) 3xf x = 2( ) 9 5g x x x= + . . gC (1,4)A .

. 2( 5 6) ( ) 4f x x g x x + = .

. : ( 1) 5 ( ) 2lim( 1) ( ) 2

x

xx

f x f xAf x f x+

+ + =

+ + +

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

x

xx

f x f xBf x f x

+ =

+ + +. f g .

. 2 1( )

( ) 1xI dx

f x

+=

+ R

55

f R ( )0 0f = ( ) ( )0

xF x f t dt=

. ( )F x R .. ( )F x .. ( ) 0F x .

. ( ) ( )3 1 1

0 01

x xf t dt x f t dt

+ ++ = + ( )0,1 .

56

. :f R .

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

. [ ], : 0,f g a R

0 0( ) ( )

a af t dt g t dt= , f [ ]0,a g [ ]0,a .

:

i. ( )( )

( ]0 , 0,

x

f t dtF x x a

x=

( )( )

0

x

g t dtG x

x=

.

ii. ( ) ( )0 0

x x

f t dt g t dt [ ]0,x a

iii. , [0, ]x y a 0 0

( ) ( )y x

x g t dt y f t dt

57

f [ ]0,e ( )0 0f = .

. ( ) ( ) ( )0

exf x dx ef e f e =

. ( )0,e ( ) ( ) ( )0

exf x dx e f e f =

. ( )1 ,e ( ) ( )( )10exf x dx ef e =

. [ ]2 0,e ( ) ( )2 20xxf x dx e f =

58

f R (1) 2f = ( ) 0f x > x R . :

. H ( ) ( )1

xg x f t dt= R (1, (1))A g .

. 1a > ( ) ( ) ( )1

0 11

aa f t dt f t dt

http://www.mathematica.gr/forum/viewtopic.php?f=54&t=22769http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=129http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=129http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=2974

1

f R , 0x ( )0

0

2limh

f x hm R

h

= .

0( ) 2mf x =

2

f : R R

(2) 2f =0

( ) lim 33x

f xx= ( ) 0f x (0, 2)x

. (0) 0f = . (0) 9f = . f (0, (0))A f . ( ) 0f x = (0, 2) . (0, 2) ( ) 2f = . 1 2, (0, 2)x x 1 2( ) ( ) 1f x f x =

3

( )3 2

1 ,6 2

xx xf x x e x R

= + + +

.

. , , . . f

. 3 2

1,6 2

x x xe x x R + + +

. :g R R ( ) ( ) ( )3

0lim 1

6g x

x

g xe g x

=

. ( )

0lim 0x

g x

=

-

50

4

f R ( ) ( )( )2

f a ff x +

x . . 1 ( , )x a 1( ) 0f x =. ( ) ( ) 0f a f = = . ( , )a ( ) 0f = .

5

f R 3 ( ) 3 ( )f x f x x+ = x R . . f

. f 1f . f . f ,

. 0 < < , ( ) ( ) ( )f f f

>

. i. ( ) ( )g x f x x=

ii. 2 2( )f x x x x + y (1) 6, (1) 3f f = = . . ( )f x 0x > . . ( )f x (0, )+ . . f . . f (0, )+ f (1, (1))A f .

. , ,a R 1a = 103

a + + = , 2 2 2 1a + + > .

7

:[0, )f R+ , ( ) ( ( )) 2f x f f x x+ = [0, )x +

. ( ) 0f x [0, )x + (0) 0f = .

. ( ) ( ) 0

( ) ( ) 1

f xe f xf x f x

+ =

(0,1) .

. f 0 1x = f (1, (1))A f .

8

( ), : 0,f g R+ , f , ( )1 1f = ( ) 0f e = , ( ) ( )f x g xe x ce = + , 0x > c R. ( ) ( ) lnx f x x x x = +

Rolle [ ]1,e. , ( ) lnf = . c . f

9

1

*( ) ,xf x xe x R= . f . f . f

. 1xe

x

= , R

. ( )

20

ln 11limx

xf

x x +

10

f R (0) 0f =

0( )0 000

( 3 ) ( 2 )lim 10 f xh

f x h f x h x eh

+ = 0x R

. f . f .

11

2( ) ( 2) 2 3 2,xf x x e x x x R= + + . f f . f . . f . . f . ( ) 0f x = f . fC

12

( ) 1xf x e x= + . . f , .

. f ( 1) 1x xe x e a = a

. 2

1 , 02

x xe x x < + > .

. 2( ) ( )f x f x lnx= +

13

, :f g R R ( ) xf x e= 2( )g x x x= . . ( )f x (0,1)A ( )g x

. ( 1,0)a 2 1 0ae a+ + = . ( ) ( ) ( )h x f x g x= , :

i. 2( ) 1h x a a x Rii. ( ) 2012h x =

14

f ( ) 2 2xf x e x ex= + + . . f . . N ( ) 2f x =

15

:f R R *a R+ , :

( ) ( ) ( ) ( )2 lim2x a

a f x x f a a f a f ax a a

=

16

:f R R :g R R ln( ( )) ( )f x g x= x R . . f R. . :

i)( (1)) ( (2)) ( (3))lim 0( (4)) ( (2)) ( (3))

x x x

x x xx

f f ff f f+

+ +=

+

ii)( (1)) ( (2)) ( (3))lim( (4)) ( (2)) ( (3))

x x x

x x xx

f f ff f f

+ +=

+

. 1

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

x

x

f f gx

=

(1) 1f (1) ln 2g = .

. (1) 2f = lim ( )x

g x+

= + , 0 (1, )x + 0( ) 2012f x =

17

f 25 5( ) 2

5

x x

f x xln

= + .

. f 0 0x = .

. N 0 0x = .

18

:f R R 3

( ) 6lim 03x

f xx

=

5

(6 30)lim (5)5x

x fx

=

.

. ( ) 2f x x= + (3,5) . . f , ( ) 0f x = (3,5) .

19

( ) ( ) ( )3 33 ln 1 , 1,xf x x x x= + + + + ( ) 5 ln 2,xg x x x x R= + .. , , . . fC gC .

. , f

( )1,0 g ( )0,1

20

. 0r > 3 0lnr r+ =

. : (0, )f R+ 1( ) (1 )( 2)f x lnxx

=

i. ( )f x .ii. r () :

.2( 1)( ) 0rf x

r

+ 0x >

. ox r> ( ) ( ) 0o of x f x+ =

21

f : R R :

( ) 0f x , x R2

2

( 2) ( ) ( 4) lim 21 1x

x f x xx

+ =

.i. (2) 5f =

ii. 0 (1, 2)x , 0 0 0

1 1 20121 2 ( )x x f x+ =

. 2 2 2( ) ( ) 2f x f x x+ = , x R , : i. (1) 2f = ii. (1, (1))fiii. (1, 2) , ( 3) ( ) ( ) 1f f + =

22

. ( ) ( )xf x ln e x x= . , ;

. ( ) ( )xg x ln e x= ( ), ( )f x g x , 0ox x= > , 'y y

. ( )f x 1 2,r r :

( )( )1( )1 1 1) ( 2 1f rf r e r r = (1) 1 2 21

22

r r rer

=

(2)

23

( )f x , [0,1] : 2 22 ( ) (0) (1) 2f x f f + + [0,1]x

.i (0,1)c ( ) (1) (0)f c f f = ii. (0) 1f = , (1) 1f =

iii. ( )f x ( ) 2f x [0,1]xiv. 1 ( ) 1f x [0,1]x

A (0,1)t , : .i. 1 (0, )r t 1( ) ( ) 1tf r f t = +

ii. 2 ( ,1)r t 2( 1) ( ) ( ) 1t f r f t = iii. ( )f x [0,1]x

24

f [ ], . ( ) ( ) 0f f = = :. : ( ) ( ) 0f f < . f

. ( )0 ,x ( ) ( )0 0f x f x

25

( )2( ) ln 1f x x x= + + .. f ' . . fC .

. ( )lim

lnxf x

x+

. ( )( ) 2 3f x x f x= + .

26

f ( )11 1 1

4 2

x x

f x ln = +

.

. f 'x x. f .

. ( ) 1

142

f xx

ln +

0x < .

27

. ,h g ( ) ( )h x g x x 0x , 0 { }x R . :

i. 0 0

lim ( ) lim ( )x x x x

g x h x

= =

ii. 0 0

lim ( ) lim ( )x x x x

h x g x

= + = +

. :f R R :

( )2

22012 xxf x

x e =

+ x R .

i. f .ii. ( ) ( ) 2011g x f x x= Riii. f

iv. ( ) ( )lim 2x

f x f x+

+

28

:f R R ( ) 0f x > x R , lim ( ) 0x

f x+

= .

: . f . ( ) 0f x > x R .

29

:f R R ( ) ( ) ( )4xyf x y f x f y+ = ,x y R .. ( )0 1f =. ( )0 0f = , :

i. f R

ii. :g R R ( ) ( )22 xg x f x= Riii. f

30

f : (0, ) R+ ( ) ln ln , 0f x x a a x a= > . 0a > ( ) 0f x 0x > a e= : . f x ee x 0x >

. x ee x= , 0x > . , x x x x + + 0x >

http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=873

31

:f R R , :

( ) ( )( )2 f x f x x+ = x R

. f

. : ( ) ( ) , ,f x f y x y x y R

. : ( )1 1,2 2

x xf x x R +

. f 1f

32

( ): 0,f R+ , :( ) ( )2 2f xx e f x= + ( )0,x +

. ( )f x ( )f x ( )0,x +. f . f

. 1f

33

. lnxe x x> > ( )0,x +. ( ), 0,x t t= +

( ) xf x e= ( ) lng x x= ,A B .i. ( )AB ( )0,t + ( ) ( )AB d t=ii. ( ) 0d t =

( )0,1iii. d ( )0 0,1t

34

f ( ) ( ) 11 3xf x ln e x= + + .

. .

. ( )limx

f x+

( )limx

f x

.

. . . fC .

35

ln( ) , 0xf x x

x= >

. f . .

. R , 0xx e x= >

. ( )( ) xxx x = (0, )2

. (0, ) + fC

( , ( ))M f y y 2010

. x e 2 2ln ( 1) ln( 1) ( )

2x xf x f x+ + < + + +

38

:[1,f 2] R , f ( )x > 0 , x R .

( )( )

1 12 2

ifif

+=

+ 1. :

. ( )1 2f = ( )2 1f =. ( )f x x= ( )1,2. ( ) ( )f x f x x = ( )1,2

39

( ): 0,f R+

( ) ( ) ( )2 ln1 xx f x x = 0x >. (1)f . fC

. , R , g ( ), 0 1

( ), 0

f x xg x

x

+ <

40

, : (0, )f g R+ 2| ( ) ( ) |f x g x x xlnx+ 1x > . . + 1 2,e e ,

1 2,e e .

. 2

20

( 5 ) ( 3 ) 1(4

i)

l mh

f x h f x h x lnxsin h x

+ + =

1(1)2

f = , ( )f x

. ( )g x (1, 2)A , ;

41

[ ] [ ]: 1, 1, 4f e , [ ]1,4 , ( )1 2f = ( ) 1f e e= + . :.i. ( )1 2, 1,x x e 1 2x x ( ) ( )1 2 0f x f x = =

ii. ( )1,e ( ) 0f =iii. ( )1,ox e ( ) ( ) ( )( )23o o o of x f x f x x =

.i. : 2e x y e+ = + fC ( )1,oc eii. ( )1 2, 1,e ( )1 2 1,e ( ) ( )1 2 1f f =

42

f [ 2, 2] , ( 2, 2) , (0) 3f = ( ) ( ) ( )f x f x f x x = [ 2, 2]x

z 2z i = . :

. H f .

. 2 2( ) 2 ( ) 3 0f x f x x + = . H ( ) ( ) 1g x f x= ( 2, 2) . . H f

. 2( ) 1 4 , [ 2,2]f x x x= + . f z

z z , x x .

http://www.mathematica.gr/forum/viewtopic.php?f=55&t=9007

43

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

ii. 1 2, ( , )x x 01 24 ( )( ) ( ) f xf x f x

.

. f ( ), , 1 2( , )x x

i. ( ) 0f ln2x x > 0

. 2

2xe x> x . ii.

2

1xx e+ =

. :f R R 2 ( )( ) f xf x e x+ = x R .

f fC 1x = .

45

:f R R : ( )( ) 1f xf x e x+ = + x R .

. ( 1)xe x + x R

. ( )2xf x x R lim ( )

xf x

=

. ( ) (1 )2xf x ln + 0x lim ( )

xf x

+= +

. ( )f x . .

. ( )f x 1( )f x

46

f R , : ( 1) 1, ( )f f > < ( 1) 1f + > + , R .

. , f 1o 3o . . , ( ) 1 ( )f x f x x = ( 1, 1) + . f R ( 1, 1) + ( ) 0f >

47

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

i. ( ) ( )2f x

h xx

= ( )0,x +

ii. f

iii. ( )g x R (1,0)

( ) ( )g x f x = x R . 21( )lim

lnxg x

x

48

f : (0, ) R+ (1) 1f = 2 ( )( ) f xf x

x = 0x > .

. 2( )f x x= 0x > . . M fC A M x x . A

(0,0)O , 2sec

o . 0t

M 3 , :

http://www.mathematica.gr/forum/viewtopic.php?f=55&t=9007

i. AM OM

ii. MOAiii. OB , B fC M x x

49

3 2( )P x ax bx cx d= + + + , , , , 0a b c d R a > 1 2 3, ,

. 2 3b ac> . . . 1 2,x x , : 1 2( ) ( ) 0P x P x + =

. 1 2,x x.

. 2( )( ) P xf x

x cx d=

+ + 2 25y x= +

1, 13x x= = 3 2( ) 2 12 13P x x x x= +

50

( ) : (0, )f x + , : 1

3 ( ) xx f x e = 0x > (1) , (1) 0f e f = =

. 1

( ) ( ) ( ) xg x xf x f x e= + (0, )+

. 1

( ) xf x xe= 0x >. ( )f x. :

i. ( )f x (2, (2))A f 1 ( ) ( )2

y e x e= +

ii. 1

2 ( 2)( )xxe x e + 0x >

http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775http://www.mathematica.gr/forum/memberlist.php?mode=viewprofile&u=775E 30 (+ 10 ) 30 60 50