Μια άσκηση την ημέρα -...

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    _____________ 2016

    http://lisari.blogspot.gr

    1

    :

    lisari.blogspot.gr

    http://lisari.blogspot.gr/http://lisari.blogspot.gr/

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    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