30 µ€±½±»·€„¹¬...

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  • thanasiskopadis.blogspot.com

    [1]

    30

    ( 2017-2018)

    -

    thanasiskopadis.blogspot.com

    blog O x , , .

  • thanasiskopadis.blogspot.com

    [2]

    30

    2017-2018

    thanasiskopadis.blogspot.com

    1

    , : f g ( ) ( )ln 1= + xf x e

    ( ) 11

    =

    +

    x

    x

    eg x

    e

    . f .

    . g .

    . ( ): 0,+ h , =h f g

    h

    . ( ) 1 2 = xh x e ,

    ( ) ( ) ( ) ( )2 3 4+ < +x x x xh e h e h e h e 0>x

    . = f

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

    ( ) ( )1 2 1 2 1 2 1 2ln 1 ln 1 1 1+ = + + = + = =x x x x x xe e e e e e x x

    f 1 1 , .

    . = g , x , x

    ( ) ( )1

    11 1 111 1 11

    = = = = =

    + + ++

    x x xx

    x x x

    x

    e e eeg x g xe e e

    e

  • thanasiskopadis.blogspot.com

    [3]

    g

    . 0>x ( )( ) ( )=h f x g x ( )1=x f x , :

    ( )( )( )( ) ( )( ) ( ) ( )( )1 1 1 = = h f f x g f x h x g f x

    1f f

    ( ) ( ) ( ) ( ) ( )0

    1ln 1 1 1 ln 1 ln 1>

    = = + + = = = = y

    x x y x y y yy f x y e e e e e x e f y e

    0>y

    ( ) ( )1 ln 1 = xf x e ,

    1g f :

    ( ) ( )1

    11

    00

    > >

    f

    g

    x xx

    f xf x

    ( )0, = +h

    ( )( ) ( )( )( )

    ( )

    ( )

    ( )

    1

    1

    ln 1

    1 1

    ln 1

    1 1 21 2

    1 1

    = = = = =

    + +

    x

    x

    ef x xx

    xf x e

    e e eg f x g f x e

    ee e

    ( ) 1 2 = xh x e ( )0, = +h

    . 0>x ( ) 2 0 = >xh x e

    h ( )0,+

    0>x : ( ) ( )3 33< < < xe h

    x x x xx x e e h e h e (1)

    0>x : ( ) ( )2 4 2 42 4< < < xe h

    x x x xx x e e h e h e (2)

  • thanasiskopadis.blogspot.com

    [4]

    (1)+(2) :

    ( ) ( ) ( ) ( )2 3 4+ < +x x x xh e h e h e h e , 0>x

    2

    , : f g , ( ) = g , :

    ( )( ) ( )= +f f x x f x , x

    ( )( )1 0 + =xf g x e x , x

    . f 1-1

    . g

    ( ) 1= + xg x e x , :

    . ( ) ( )( )ln=h x g x

    . g

    ( )21 1 2 2 + + =xg e x

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

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

    (2)-(1) :

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

    f 1-1

    . 0=x :

    ( )( ) ( ) ( ):1 1

    0 0 0 0

    = =f

    f f f f

  • thanasiskopadis.blogspot.com

    [5]

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

    1 0 1 0 1 0

    + = + = + =f

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

    ( ) 1 = + xg x e x

    . ( ) 0>g x

    ( ) 1 0 = + >xg x e , g .

    ( ) ( ) ( ):

    0 0 0> > >g

    g x g x g x

    ( )0, = +h

    . g , 1-1, .

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

    1 1 2 1 2 1 22 2 1 1 2

    + + ++ = + = + + =g

    x x xg e x e x g e x g

    ( ) ( ):1 1

    2 2 21 2 1 2 1 1

    + = + = = = g

    g x g x x x

    3

    ( ) =f x x x

    . f

    . f

    . f 1f

    . ( ), x y 1fC , 0>x , 5

    ,02

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

  • thanasiskopadis.blogspot.com

    [6]

    0

  • thanasiskopadis.blogspot.com

    [7]

    ( )1 ,0 = f ,

    ( ) ( ) ( )( ) ( )10

    lim , lim ,0

    = = x x

    f f x f x

    [ )2 0, = + f ,

    ( ) ( ) ( )) [ )2 0 , lim 0,+ = = + xf f f x ( ) ( ) ( )1 2 = =f f f

    . f , 1-1,

    .

    ( )1 ,0 = :

    ( ) ( )1

    2 2 1

    = = = = = x

    y f x y x x y x y f y y

    ( ) ( )1 1 ,0 = = f f

    [ )2 0, = + :

    ( ) ( )2

    2 2 1

    = = = = =x

    y f x y x x y x y f y y

    ( ) [ )1 2 0, = = +f f

  • thanasiskopadis.blogspot.com

    [8]

    ( )1 , 0

    , 0

    x ( )1 =f x x , ( ), x x

    ( ) ( )2

    225 250 4

    2 4 = + = +

    x x x x

    ( ) 2 2544

    = +g x x x .

    g ( ) 2=x

    ( )2, 2 .

    4

    ( )2

    3

    2 1 , 0

    , 0

    + + =

    + >

    x x xf x

    x x a x

    . , f 0 0=x

    1=

    . ( )limx

    f x ( )lim+x

    f x

    . ( ) 0=f x 1 2,x x 1 20< x f

    2

    3

    1lim

    1+=

    +x x x x

    .

  • thanasiskopadis.blogspot.com

    [9]

    ( ) ( )20 0

    lim lim 2 1 1

    = + + =x x

    f x x x ,

    ( ) ( )30 0

    lim lim+ +

    = + =x x

    f x x x

    ( )0 1=f

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

    lim lim 0 +

    = =x x

    f x f x f ,

    1=

    1= ( )2

    3

    2 1 , 0

    1 , 0

    + + =

    + >

    x x xf x

    x x x

    . ( ) ( )2 21lim lim 2 1 lim 2 1

    = + + = + + + = x x x

    f x x x x xx

    2

    1 1lim 1 2 1

    + + + =

    xx

    x x

    ( ) ( ) ( )3 3lim lim 1 lim+ + +

    = + = =x x x

    f x x x x

    . ( )lim

    = x

    f x 3 0

  • thanasiskopadis.blogspot.com

    [10]

    Bolzano ( ) 0=f x ( ) ( )2 40, 0, +x x

    ( ) 0=f x 1 2,x x 1 20< x , ( ) 3 1= +f x x x

    , f .

    ( )21

    lim+x x f x

    , 2 0>x

    ( ) ( )2

    2lim 0+= =

    x xf x f x , f 2x

    ( ) 0=f x 0>x

    ( ) ( ) ( )2 2 0> < + > >f

    x x f x f x f x g

    ( ) ( )1 2 0 x

    . ( ) 2x

    . ( ) ( )1= +g x f xx

    , 1>x

  • thanasiskopadis.blogspot.com

    [13]

    i. g

    ii. 0 1>x , 01

    0=xe x

    iii. ( ) 3= g x x , 1>x

    . ( ) 2 0 >f e , ( )( ) ( )( )

    ( )( )( )3 3

    2 2

    2 1 2lim lim lim 2

    2 +

    = = = +x x x

    f e x x f e xf e x

    x x

    ( ) 2 0 x ( )2 2ln 0= h x x , 1>x

  • thanasiskopadis.blogspot.com

    [14]

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

    0 =x e :

    ( ) ( )( ) ( ) ( )( )22 ln 1 2 0 = = f e f e e f e f e

    ( ) 0=f e ( ) 2=f e

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

    ( )1, = + , g ,

    ( ) ( ) ( )( ) ( )1

    lim , lim ,2++

    = = x x

    g g x g x

    ii. :

    ( )1 1 1 1

    ln ln ln ln 0 1= = = = =x xe x e x x x g xx x

    ( )1,+

    ( )1 g g , ( ) 1=g x ( )1,+

    0 1>x , 01

    0=xe x

    iii. :

  • thanasiskopadis.blogspot.com

    [15]

    1 1 3 2 x x x

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

    , 1>x , ( ) 3= g x x , ( ) 3= g x x , 1>x

    7

    = , 3 = 2 = , ( )0, +

    . x , x,

    ( )2

    2

    , 0

    2 , 3

    < =

    <

    x xf x

    x a x

    . f ( )0, +

    . f ( )0, +

    1=

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

    . f 1f

    . 1f

  • thanasiskopadis.blogspot.com

    [16]

    . 0< x

    ,

    22

    = =

    xx

    ( ) ( )( ) 21 1 22 2

    = = =x x x x

    3< a x

    ( ) ( ) ( ) ( )( ) ( )( ) ( )1 1 2 22 2

    = + = + = + =x x

    ( ) 2 2 21 2 2 2 2 22

    + = + = x x x

    ( )2

    2

    , 0

    2 , 3

    < =

    <

    x xf x

    x a x

    . 0<

  • thanasiskopadis.blogspot.com

    [17]

    ( ) ( ) ( )2 2

    lim lim lim 2

    = = + =

    x x xf x f x

    xx x

    ,

    ( ) ( ) 2 2 22 2 2lim lim lim 2

    +

    = = =

    x x xf x f x x

    x x x

    f =x

    f ( )0, +

    1= ( )2 , 0 1

    2 1 , 1 3

    < =

    <

    x xf x

    x x

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

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

    1 2 1 2 1 22 2

    = = =x x

    f x f x x x x x

    , 1 2x x . .

    . ( ]1 0,1 = f , ( ) 2 0 = >f x x 1-1 ( ) ( ]1 0,1 =f

    ( ]2 1,3 = f , ( ) 2 0 = >f x 1-1 ( ) ( ]2 1,5 =f

    ( ) ( )1 2 =f