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[1]

30

( 2017-2018)

-

blog O x , , .

[2]

30

2017-2018

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

[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)

[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

[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

[6]

0

[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

[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

.

[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

[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

[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

[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. :

[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

[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<

[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