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:

lisari.blogspot.gr

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http://lisari.blogspot.gr

2

2016

27

(1/4/2016) 10/4/2016

G : 0, R :

x

G(x) G(x) 3x , x (0, )2

G(1) 0

G'(1) 1

.

3x lnx ,x 0

G(x)0 ,x 0

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B. i. N G

ii. 3

3G(x) xG'(x) x ,x 0

. 31 3lnx x lnx ln lnx .

. B

G .

G : R R 2

:

x

G(x) G(x) 3x , x (0, )2

G(1) 0 G'(1) 1

. G R .

B. i. () ,

G

ii . () ()

xx 3

Cf 2 .

. B G ,

() yy .

.

31 3lnx x lnx ln lnx

.

.

1 1

3e G 3eGxc 3 c 2

1 1G(t)dt G(t)dt xe x 1 0 ,c 0

0,1

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1 ( )

. x > 0

2

2

4

x x G''( x ) 2xG'( x ) 3G'( x ) G''( x ) 3x 2G'( x ) xG''( x ) 3x

2 x x

2 2

G'( x ) G'( x )' 3lnx ' 3lnx c ,c R

x x, G'(1)

3ln1 c c 11

2 2 2 32

G'( x ) 13lnx 1 G'( x ) 3x lnx x G'( x ) 3x lnx x

x x

3 3G'( x ) x lnx ' G( x ) x lnx k , k R , G(1) k k 0 3G( x ) x lnx ,x 0 ,

3

x 0 x 0 x 0 DLH

3

lnxG(0 ) limG( x ) lim x lnx lim 0

1

x

.

3x lnx ,x 0G( x )

0 ,x 0

B. i.

2 23x lnx x ,x 0G'( x )

0 ,x 0

ii.

2 2 3 3 3

3

G'( x ) 3x lnx x xG'( x ) 3x lnx x xG'( x ) 3G( x ) x

3G( x ) xG'( x ) x ,x 0

G(0)=0 , 33G( x ) xG'( x ) x ,x 0

.

G 1, 2 2G'( x ) 3x lnx x 0 G G 1, 0 ,

x 0 lnx 0 x 1

lnx x 1 x 1

x

0

3

1

e

G

- +

G

1

3e

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3 3 3 3 3

3 3

1 3lnx x lnx ln lnx 1 lnx x lnx ln lnx ln lnx lnx x lnx 1

ln x lnx x lnx 1 lnG( x ) G( x ) 1 G( x ) 1

G G 1, 0 ,

1, .

.

3x lnx ,x 0G( x ) 0 0 x 0 x 1

0 ,x 0

xx CG 0 , 1 . G

3

1 1G 0

3ee

G x 0 ,x 0 ,1 1

0

E G( x )dx .

B.ii. 33G( x ) xG'( x ) x

1 1 1 1 1

13 3

00 0 0 0 0

3G( x ) dx xG'( x ) x dx 3 G( x ) dx xG( x ) G( x ) dx x dx

1 1 13 G(1) E 4E E ..

4 4 16

A.

x > 0

2

2

4

x x G''( x ) 2xG'( x ) 3G'( x ) G''( x ) 3x 2G'( x ) xG''( x ) 3x

2 x x

2 2

G'( x ) G'( x )' 3lnx ' 3lnx c ,c R

x x, G'(1)

3ln1 c c 11

2 2 2 32

G'( x ) 13lnx 1 G'( x ) 3x lnx x G'( x ) 3x lnx x

x x

3 3G'( x ) x lnx ' G( x ) x lnx k , k R , G(1) k k 0 3G( x ) x lnx ,x 0 , (1) G R , G( x ) G( x ) ,x R

x 0 x 0 , ( 1)

3 3G( x ) G( x ) ( x ) ln( x ) x ln( x )

G(0 ) 0 G R

3x ln x ,x 0G( x )

0 ,x 0

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32

x 0 x 0 x 0 x 0 x 0 DLH

2

G( x ) G(0 ) G( x ) x lnx lnxG'(0 ) lim lim lim lim x lnx lim 0

1x x x

x

2 23x ln x x ,x 0G'( x )

0 ,x 0,

B. i.

x , 0 . : y = CG

() G( x ) x ,x R . x=0

, (0,0) . R* .

2G( x )

G( x ) x ,x R* ,x R* x lnx , x R*x

( 2)

2(x) x ln x ,x R* '(x) 2x ln x x x 2ln x 1 (2) (x)= .

1 1

'(x) 0 x 2ln x 1 0 ln x x2 e

H

1 1 1

2ee e

1 1 1 0, 0, ,0

2ee e

2

x 0 x 0 x 0

2 3

1lnx xlim x lnx lim lim 01 2

x x

1 1 1 , , ,

2ee e

2

xlim x ln x

x

-

3

1

e

0

3

1

e

G

+ - - -

G

1

3e

1

3e

x

-

1

e

0

1

e

H

- + - +

H

1

2e

1

2e

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1

2e

(x)= G( x ) x ,x R

x=0

1

2e

(x)= 2 G( x ) x ,x R 3

( x=0)

1

0 2e

(x)= 4 G( x ) x ,x R 5

( x=0)

0 (x)= 2 G( x ) x ,x R 3

( x=0).

B. ii.

.i. CG 3 ()

xx (

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2 2 2

1 1 1 1 5E

16e 4e 16e8e e 4e e ..

.

G 1, 2 2G'( x ) 3x lnx x 0 G G 1, 0 , x 0 lnx 0 x 1

lnx x 1 x 1

3 3 3 3 3

3 3

1 3lnx x lnx ln lnx 1 lnx x lnx ln lnx ln lnx lnx x lnx 1

ln x lnx x lnx 1 lnG( x ) G( x ) 1 G( x ) 1

G G 1, 0 ,

1, .

.

1 13e G 3eG

c 3 c 2 x

1 1

F( x ) G(t )dt G(t)dt xe x 1 ,x 0 ,1

0,1

21 1

3eG 3eGc 3 c 2

1 1

F(0 ) F(1) G(t )dt G(t)dt 0

1 1 1 1

3eG 3eG 3eG 3eGc 3 c 2 c 3 c 2

1 1 1 1

G(t )dt G(t)dt G(t )dt G(t)dt

1

3eGc 3

13eG

c 2

G(t )dt 0 ,

1 1 3eG , 3eG

c 2 c 3

G(t )dt 0 3

1 1 1 10 1 1 3e G 0 , 1 3eG 0

c 3 c 2 c 2 c 3

G

G( t )dt 0

G(t )dt 0 G(t )dt G(t )dt 0 G(t )dt 0

-1

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1 12 3

c 2 c 3 ,

(3)

1

3eG3

13eG

2

G(t )dt 0

F(0 )F(1) 0 .Bolzano 0,1 F . .

2 ( )

A. :

2 2 3 2 2( ) 2 ( ) 3 ( ) ( )

2 ( ) ( ) 3 3ln 3lnG x G x G x G x

G x xG x x x x cx x x x x

x=1: c=1 :

2 2 3 32( )

3ln 1 ( ) 3 ln ( ) ln ( ) lnG x

x G x x x x G x x x G x x x cx

x=1: c=0 : 3( ) lnG x x x , 0x . (1)

G 0 :

3

3

0 0 0 0

3

ln(0) lim ( ) lim ln lim lim 0

1 3x x x x

x xG G x x x

x

.

( .. : ( ) ( )G x G x .

(1) x 0x : 3 3 3( ) ( ) ln( ) ( ) ln( ) ( ) ln( ), 0G x x x G x x x G x x x x )

. i) GC .

0, . 0x

1

2 2 2 3( ) 3 ln 3ln 1 0G x x x x x x x e

.

1

3( ) 0G x x e

G 1

3 ,e

G 1

30,e

.

GC ..

1

3e

1

31

3G e

e

0 0,

0, . 1

3e ..

1

3e..

1

3e

.

ii) E De L H. (0) 0G x=0.

0x 3 2 2 3 3( ) 3 ln 3 ln 3 ( )xG x x x x x x x x x G x . . 1x :

3 3 3 31 ln ln(ln ) ln 0 1 ln( ln ) ln 0 ln ( ) ( ) 1 0x x x x x x x x G x G x (1)

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( ) ln 1h x x x , 0x . ln 1, 0x x x

1.

(1) ( ( )) 0 ( ) 1h G x G x G1

,3e

lim ( )x

G x

, 0 1x : 0( ) 1G x (G

1

3 ,e