Σπιράλ 6_Παράγωγοι Γ' (2012-13).pdf
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Transcript of Σπιράλ 6_Παράγωγοι Γ' (2012-13).pdf
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555 1
2.6 )
1. , :
( ,) , ( ,] , (, ) , [, ) , [,], [,), (,], (,) ( , ) .
(..1
f(x)x
).
2. . .. f , -
f (x) 0 , x .
, , f (x) 0 , x .
3. , f .
f.
4. , -
.
[]
1. f (,)
*1 2 k
x ,x ,...,x (,) (k ) , f
( (,) ) f
1 1 2 k(,x ) (x ,x ) ... (x ,) , f (,) .
[ ]
1. :
) .
) .
) ( ).
) -
.
23
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. , f
.
2. (
),
( g(x) ) f (
g) f.
, f (x) f (0) 2 0 , x ,
f
f x 0 , x ,
f . 0
x 0f x 0
0f x f x 0 0x x 0f x f x 0 0x x .
3. - :
f (x) 0 .
f (x) 0 .
f (x) 0
.
4. ( ).
) . -
,
.
.
) ,
:
f [,],
[f(),f()]. ( [f(),f()] )
f (,), -
x x limf(x),limf(x) . ( x x limf(x),limf(x) ) ,
.
) f -
(,),
.
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) f
(, ), f
(,) .
) f
(,), .
) ! , , -
2 .
5. , -
.
6. f(x) g(x) f(x) g(x) , -
h(x) f(x) g(x) .
h , .. , 0 h
x
0
h(x ) 0 . ,0
h(x) h(x ) 0 0
x x , 0
h(x) h(x ) 0 0
x x .
h 0 h
x , -
.
.
. f
f g(x) f (x) g(x) (x) .
[ ] [2,3,4, 3,4. .256-7]
1. :
)3x
f(x) 2x x3
, x
) 2f(x) x x 1
) xf(x) 1 x
) xf(x) x e (x x 1) 2x , x [ ,]
) 2
f(x) x 2x, x [ , ]2 2
2. 1
f(x) x 1x 1
. N :
) x 1 x 1lim f(x), lim f(x)
) f.
[0,1) (1,2] ;
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[ - - f] [5, 5. .256-7]
3. 2x 7
f(x) 4x 4 ln x2 2
, :
) f .
) f(x) 0 .
[ ] [6. .257]
4. x
2 2
ef(x)
x
.
[ f ]
5. 2f(x) 2x ln x x 1 .
) f, f (x) f (x) .
) f f.
) 2x 1
2lnxx
.
) 0 x 1 , 2
xln x 1
2x 1
.
[ f]
6. x
f(x)x
0,
2
.
[ f f]
7. N x 21 e x x
f(x) ln , x 02 8 2
.
[ ] [A5,6, B2, 5, 257]8. N :
) x ln x 1
) 2x 2x 3ln x 3
) x 2e 1 ln x 3
) x x x x2 3 4 9
9. f [1,e] 0 f(x) 1 f (x) 0
x [1,e] , 0x (1 ,e)
0 0 0 0f(x ) x ln x x .
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[ ]
:
ln x x 1, x 0 (1), .
[ 2, 266 ( )]
xe x 1, x (2), (1) x xe 0
ln x x 1 x , ln x x, x 0 , (1)
xe x 1 x , xe x 1 x , (2)
10. :
) x ln x x 1 , x 0
)2
x xe x 12
, x 0
) i) x 2x, x 0 ii)3x
x x , x 03
) 0 1 e 1
11. 2x 2x 1
2
2x 1ln e e
x .
12. g : .
) 2g(x 2x) g(x 2) .
) 2x 2x x 2 2 x 3x 2, 1 .
;
[ ]
13. ln x
f xx ln x
.
14. ln x
f x 1 xx
15. ln x
f xln x 1
.
16. f : :2 x2xf(x) (x 1)f (x) e , x , f(0) 1 .
) x
2
ef(x)
x 1
, x .
) f.
17. f 2 xf(x) ln(1 x e ) - .
18. 2f(x) x 2 (1 x)(ln x 2) .
) f .
) 2
x 2 (1 x)(ln x 2) .) f.
) 2(1 x)(ln x 2) x 1 , x 0 .
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19. :
) 2ln x 1 x x 6 0
) 2ln x x e 0
20. f, g . f g 1 1 .
) g 1 1 .
) : 3g(f(x) x x) g(f(x) 2x 1)
.
21. f, g , f(1) g(1) .
) f (x) g (x) , x , .. f(x) g(x) ( ,1) f(x) g(x) (1, ) .
) f (x)g (x) 0 , x , .. f(x) g(x) .
22. ln x
f(x) , x (1, )x 1
.
) (1, ) .
) , (1, ) , .
23. f , :
f (x) x f (x) 3f (x) , x . f -
.
24. f : (0, ) ,
f(x)f (x) 2lnx, f(1) 2
x .
) f .
) f .
25. f xf(x) e x .
) f .
) :2x 2 x 2e e x x 2 .
26. f : f(x) 0 x
3 f(x) 3 2f (x) ln f(x) e x x 2x 1 x .
) f .
) 2
f(lnx) f(1 x ) .
27. 1 1
ln 1x x 1
, x ( , 1) (0, ) .
28. ) x
31f x x2
.
) : 2 2 3x 2 x x x 2 62 2 2 [x x 2 ]
29. xg (x) x g(x) x , : x
g x
x
x 0 .
30. f : f (x) 2f(x) f(0) 1 . -
2xf(x) e x 0 .
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2.7 Fermat
1. .
2. . .
3. ,
( ).
4. ,
. [;]
5. , , ,
, , -
.
6. [ (. 195)]
[,], f [,]
m. , 1 2
x ,x [,]
1m f(x )
2M f(x ) : m f(x) M x [,] .
7. Fermat : -
f 0 0
A(x , f(x )) xx.
x1 x2 x3 x4
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8. Fermat .
0
f (x ) 0 ,
f 0
x .
.. 3f(x) x , f (0) 0 ,
x 0 .
x0 -
9. , 0
x f
C ,
f -
0
x .
10. , , f -
. .. f(x) x x 0 ,
.
11. 0
x , f 0
x
0x ,
0f (x ) 0 .
.. 2f(x) x , x [0,2] , 0
x 2 , -
f (2) 4 0 .
12. , Fermat.
13. x f (x) 0 , f ( ).
14. f 0 0
(,x ) (x ,) 0
x ,
f .
15. 0
x f , f -
0x , f 0x f (
0x )
16. f 0
x
f ,
f 0
x 0
x .
17. f :
) f(x) 0 , 0 .
) f(x) 0 , 0 .
O
=x3
x0
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1. ( ):
f(x) 0 ( f(x) 0 ), f
0 ( 0 ).
f(x) g(x) ( f(x) g(x) ) x,
h(x) f(x) g(x)
. h
0
x h() 0 , h(x) h() h(x) 0
f(x) g(x) x .
f(x) x ,
f,
f(x) x ,
f .
2.
f(x) g(x) (1).
(x) f(x) g(x) .
g(x) 0 x A , f(x)
(1) 1g(x)
-
f(x)
(x) 1
g(x)
.
f(x) 0 g(x) 0 x A , (1) ln f(x) ln g(x)
(x) ln f(x) ln g(x) .
-
- .
3. Fermat:
f , :
0x f , -
f, x
0
x 0
f (x ) 0 ,
f .
f(x) (f(x) ),
0x (,)
0f(x ) ,
0f(x) f(x ) (
0f(x) f(x ) ). f(x) ( )
0x ,
Fermat0
f (x ) 0 .
-
.
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f -
1 2 x , x , ,x ,
1 2 f x f x f x 0 . -
.
4. :
( x) , -
, .
.
5. :
f , -
, .
:
0
x :0
f (x ) 0 .
M c . f(x) c x .
[ ] [1,2,3,4, 6* 267-9]
1. :) 3 2f(x) x 4
)ln x
f(x)x
2. xf(x) e x, 0 . -
f .
[ - Fermat] [5 268]
3. 0 x 0 xlnx x , .4. x x x x xf(x) 2 5 7 10 . 0 ,
f(x) 1, x .
5. f : x 2f(x) e ln(x 1), x .
fC A(0,1) .
[ f ] [4 269]
6. f : , :3 2 x 2f (x) f (x) f(x) e x x 1 , x .
) f .
) f .
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7. f [,].
) f f(), f () 0 .
) f f(), f () 0 .
) f ()>0 f ()
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18. 2f(x) xln x . Cf f
.
19. x 2f(x) e x x 1 .
) Cf .
) x 2e x x 1 .
) xe 1 x(1 x) , x .
20. i) f 2x
f(x) x lnx 2
-
.
ii) ,
2
2 2
ln
.
21. f(x) ln(lnx) lnx .
) f .
) : ln(lnx) lnx 1 .22. f : f (x) f (x)
x . f 0
x 0 f(0) 0 :
) x 0 f(x) f (x) .
) x 0 f(x) f (x) .
23. f : 2 2f(x) x 1 2xf(x) ,
x . f .
24. A. f [,] :f () 0 , f () 0 . :
) (,) f .
) f () 0 .
B. ( Darboux) f [,]
f () f () 0 , (,) , f 0 .
. ( Darboux) f [,]
f () f () 0
x (,) , 0
f (x ) .
. f f (x) 0 x f
.
[:
1) f 1-1 .
2) f (x) 0 x f f -
1-1 .]
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2.8 -
[ ]
1. :
) f , f
C
0x
fC ( ). :
x 0 0 0f(x) f(x ) f (x )(x x ) . [ ]
) f , f
C
0x
fC ( ). :
x 0 0 0f(x) f(x ) f (x )(x x ) . [ ]
2. :
) f f
C xx 0
x ,
f(x) 0 x .
) f f
C xx 0
x ,
f(x) 0 x .
3. f
C .
[ ]
4. .
5. 0 0
f (x ) f (x ) 0 , 0
x
. ,
f , f
.
6.
,
.
7. ( ) .
: [ ]
) f ( 2 ) , f (x) 0 , x .
) f ( 2 ) , f (x) 0 , x .
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8. f
, -
f ,
(.x. 4f(x) (x 1) ).
9. 0
f (x ) 0 x , f .
[]1. [ Jensen]
) f [,] .
: f f
f2 2
.
) f [,] .
: f f
f2 2
.
2. [ Jensen]
f :
[,],
[,], [,]
(,), ,
1 2
x ,x (,) 1 2
x x , *1 2 , 1 2 1 , :
1 1 2 2 1 1 2 2f( x x ) f(x ) f(x )
1. f ,
f , f (x)
f (x) 0 f (x) 0 .
f (x) 0 , .
f (x) 0 , .
f (x) 0 ,
.
2. f , -
f (x) 0 f .
f . -
.
3. f ( -
) ( ) ,
f (x) 0 ( f (x) 0 ),
, 0
x
0f (x ) 0 .
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4. f
1 2
x , x , ,x , 1 2
f (x ) f (x ) f (x ) 0 .
.
5. f ,
) 0x f , f, -
x 0
x 0
f (x ) 0 .
) f -
.
6. :
) Jensen [ ]
) : f -
f. -
f(x). -
Cf . f
.
[ ] [1-4, 1-5 267-9]
1. f : 0
x f.
) f , 0
f(x ) f.
) f , 0
f(x ) f.
2. f : . 0 0x ,f(x )
0
x .
3. ) 6 4f x x 3x 5( x) .
) f
C f
C .
4. x 2
f (x) 2e x , 0 .
5. 3 -
.
6. f xf (x) 2x 0 x .
A 0, f(0) fC .7. 5 4 3 2f(x) x 5x 10x x , x , , ,
2 .
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[ ]
8. f(x) ln(ln x ) .
) f .
) f
C 0
x e .
) x
ln(ln x) 1 0e
, x 1 .
9. f : f (x) 4f(x) 4f (x) x
2x
f(x)g(x) , x
e .
) g .
) g
C x x , f(x) 0 x .
10. i) f(x) lnx x .
ii) 2 2g(x) ln x 2xlnx x 3 .
iii) gC 0x 1
.i) x lnx 4x 3 x 1 .
11. f g . :
) f g
C ,C .
) f
C g
C , f
C
g
C .
12. f : (0, ) f(x) x x
f (x)
x f(x)
x 0 . :
) f .
) f (0, ) .
[ Jensen]
13. 0 , 1 : (e 2) (e 2) 2 3 .
14. f :
. -
x :3x
3f(x) 4f4
.
15. f f(x) ln(ln x ) .
) f .
) 1 1 , : 2
ln ln ln2
.
16. f f(2) 0
[ 2,5] , : 4f(5) 3f( 2) 0 .
17. f : [0, ) . f(0) 0 ,
x f(x)
f2 2
x 0 .
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2.9) DE L HOSPITAL
1. De L Hospital :
) 0x x
f(x)lim
g(x)
0
0
) f , g
0 0 0 0(x ,x ) (x ,x )
0 0(x ,x )
0 0(x ,x ) ( ,) (, ) .
) f , g 0x , -
0x .
) 0x x
f (x)lim
g(x)
( ).
2. ! 0
x , -
(), .
3. De L Hospital -
, f , g :
) 0
(x ,), 0x x ( ),
) 0
(,x ), 0x x ( ).
4. 0 0x x x x
f(x) f (x)lim lim
g(x) g (x)
.
.
0x x
f (x)lim
g(x)
( ),
0x x
f(x)lim
g(x).
0x x
f (x)lim
g(x)
,
0x x
f(x)lim
g(x) ,
.
5. De L Hospital ,
-
.
6.
0
0
.
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7. De L Hospital .
8. 0
0
, -
.
9. De L Hospital
, .
-
[0
0: 1 ] [1, 3,4,5 285-6]
1. :
)4
2x 0
xlim
x +2x-2, )
2
xx 1
ln xlim
e ex , )
2x 0x
x-xlim
xe -1-x-
2
, )
2
x 1
x 1lim
ln x
[ : 2 ]
2. :
)x
ln( x 1)lim
ln( x 2)
, )
x
x
elim
x, )
2
xx
x x xlim
e x 2
, )
3 2
2xx 0
(x +x +x+1)lnxlim
e
[ 0 ]
f
f g1
g
g
f g1
f
0
0
, .
3. : ) 2x 0lim x ln 3x
, )
2
1
x2x 0
1lim e
x
, )x 0lim(xln x)
.
[ ( ) ( ) ]
gf g f 1f
1 1
g f1 1f g1 1 1 1
f g f g
.
: [ 0 ( ) ( ) ]
( ):
) , ,
) ( ) , .
4. :
)x 1
1 1lim
ln x x 1
) 2 x
xlim(x ln x e )
)
2 x
x 0
x elim -
x x
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[ 0 00 ,0 , ,1 ]
gf .
:g(x) g (x ) l n f ( x)
f(x) e f(x) 0 .
5. :
) xx 0limx
, ) x
x 0lim(x )
, ) x
x 0lim(1+x)
, )
1
x
xlim(x 2)
, ) x
x 0lim(x)
, )
1
x
x 0lim(1+x)
[ ] [3 286]
6. ,, 3 1-x
2x 1
x +e +x+4A lim
x -2x+1
.
[ ]
7. f 0 f(0)=0 f (0)=1 :
) x 0f(x)
lim x , ) xx 0xf(x)
limx-xe
8. f : :5 3f (x) f (x) f(x) x x x ,
) x 0limf(x)
,
) 3x 0
f(x) 1lim
6x .
9. * A f: :
)h 0
f (x-2h)-f (x)lim 2f (x)h
,
)2h 0
f (x+h)-4f (x-2h)+3f(x-3h)lim 6f (x)
h .
***
10. 2 1x ,x 0
f x x
0 ,x 0
.
) f .
) f 0
x 0 .
) f . :
0
0
x x x
f x f f xf lim lim lim f x
x 1
. H f -
f .
. .
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2.9)
1. f -
.
2. , .
3. Cf ( -
).
4. :) ( ) xf x e ( y=0).
) ( ) f x lnx ( x=0).
) ( )
f xx
( y=0 x=0).
) ( ) f x x ( x= +
2, ).
) ( ) f x x ( x= , ).
5. :) 2.
) 2.
) x x .
6. f(x) c f(x) x .
7. y x f
C ( )
x
f(x)lim
x
xlim[f(x) x]
.
8. x
f(x)lim
x ,
fC !
[ f
C ]
9. -
( ) /
.
.
, ,
!
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f
y=, -
, -
, , , f
C
.
,
( ) -
.
:
21x x , x 0f(x) x
0, x 0
, -
y x .
:
x
f(x)lim ... 1 0 1
x
x lim(f(x) x) ... 0
, f(x) x :
22 2 2 2
2
x 2k 01x 0 x 0 x 0 ,k x k,k
x x 2k ( 0)
x k ,k
!
, :
10. -
.
11. f 0
x
0x x , . f :
1,x 0
f x x2 ,x=0
.
, f .
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1. :
)
)
2. [, , -
(, ) , ( ,) ].3. ,
.
4. f
C , -
.
5. x
f(x)lim
x .
. .
[1-3, 1,2 285-6]
1. x
f xx
xx .
2. f : :3 2
2
2x 3x 12x 3 f(x)
x
, *x .
f
C .
3. ,, f 2( 1)x x 5
f(x)3x
x 2 y 3 .
4. 2x y 5 0
f . :
(i)x
f(x)lim
x(ii)
xlim f(x) 2x
(iii) 2x
f(x) 3x xlim
x f(x) 2x 3
5. f : . f
C :
) ,
) .
6. f : -
() : y x . f
C , :
) f ,
) f .
7. y 2x 1 f
C f -
y x 1 fC g.)
xlim f(x)
.
) g f .
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2.5 - 2.6 . Rolle, ..., ... -
-
.Rolle
1. f [,] f (x) 0 x [,] ,
f 1-1 [,] .
2. f [,] f() f() , -
0
x (,) 0
f (x ) 0 f (,) .
3. f [,] f() f() ,
0
x (,) f
C 0 0
A(x , f(x )) -
xx.
4. f : [,] [,] Rolle
[,], g(x) (f f)(x) -
Rolle [,] .
5. A [,] Rolle
Bolzano f .6. f , g [,] f() g()
f() g() , 0
x (,) 0 0
A(x , f(x )) 0 0
B(x ,g(x )) -
.
7. f , f -
.
8. f [,] f (x) 0
x (,) , f(x) 0 [,] .
9. f f(2004) f(2013) f(2012) ,
0
x (2004, 2012) 0
f (x ) 0 .
10. f . A(,f())
B( ,g ()) f .
11. f . f() f() , , ,
fC xx.
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12. f [,] f
[,] .
13. f [,] f() f()
f () f ()
.
14. (,) f () 0 f() f()
0
.
15. ... Rolle.
16. f (,) [,] f() f() f () 0
(,) .
17. f() f() 0 f() 0 f
, 1 2
, (, ) 1 2
f ( ), f ( ) .
18. f [,]
, (,)
f() f()
f ()
.
19. f(x) x
[ 3,2] .
20. f [,] [,],
f
C A(,f())
B( ,g ()) .
21. f f (x) 0 x ( ,0) (0, ) , f
.
22. f f (x) 0 x , f .
23. f, g f (x) g (x) 0 x f g
C , C -
, f g .
24. f (x) f(x)
x , c :x
f(x) e c x .25. f(1) 1 f (x) f(x) x , : x 1f(x) e .
26. x , : f(x) g(x) f (x) g (x) .
27. f (2x) x x x , c : 2f(2x) 2x x c
x .
28. (x 1) f (x) 2 ln(x 1) x 1 , c : 2f(x) ln (x 1) c .
29. f (x) g (x) x , 1 2
c ,c :1 2
(f g)(x) c x c
x
.30. x 4f (e ) 5x x , c : x 5f(e ) x c x
31. f x 2 x x 0 , c : 2f(x) x c x .
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: 3 2012-2013
-3-
( )
32. f [,] (,)
f() f() f() (,) , 0
x (,) :
. (4 ) 0f (x ) 0
.0
f (x ) 0
.0
f (x ) 0
.
33. f f (x) 0 x ,
f(x) 0 :
. .
. .
. .
. .
34. -
f x [,]
. - f [,] ...
:
. .
. .
. x (,) .
. f() f() .
. .
35. f (,), [,] f
(,) ... f [,] :
. (,) f() f()
f ()
.
. (,) f() f()
f ()
.
. (,) f() f()
f ()
.
. (,)
f() f()
f ()
.
. (,) f() f()
f ()
.
36. 23f(x) (x 1)
[0,9] :
. [0,9] .
. f(0) f(9) .
. (0,9) .
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: 3 2012-2013
-1-
&
()
2:
555 1
2.7 - 2.9 , , ,
-
1. f [, ] f()f() 0 x 0,3 3,5 f
[0,5].
5. f [, ]
f() f() f [, ].
6. f [, ] f [, ]
f(x) = 0 (, ).
7. f [, ] f() f()
f() f() f() f().
8. [, ] .
9. f [, ] f () < f () m -
f [, ] [f(), f()] [m, ].
10. f [, ] m M
f [, ] f()=m f()=.
11. .
12. .
13. f, -
f.
14. -
.
15. f .
16. .
17. f 0
x f(x0)=0
x0 f .
30
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: 3 2012-2013
-2-
18. (x0, f(x0)) Cf f.
19. T .
20. f.
21. , f (x) 0
x .
22. f -
f .
23. f (, ) f (x) 0 x ,
f [, ].
24. .
25. f =(,) -
, f .
26. 3 .
27. 4 .
28. f (, ). f (, )
(, ) (, f ()) Cf.
29.
.
30. f (x) 0
x .
31. 0 0 x , f x Cf 0f x 0 .
32. f.
33. .
34. f .
35. x limf(x)
y = f.
36. 0
x x f
0x xlim f(x)
.
37. y = Cf x x
f(x)lim 0x
.
38. y = x + Cf x x
f(x)lim
x
xlim f(x) x
.
39. y = x + Cf x
2x
f(x) 1 lim
x xf(x) 1
.
40. 0xlim f(x) x 0x x f.
41. .
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42. 0x x
lim f(x)
0x x
lim f(x)
f
0x .
43. f -
f.
44. xlim f(x)
xlim f(x)
-
.
( )
45. f : 0, R f(x)e f(x) x lnx x>0.
:
: f (0, +).
: f
: f (0, +) : f
:
46. f f 3 . f :
:
:
:
:
:
47. f (,), :
: [, ] f()f()=0.
: (,) f()=f().
: 0x , 0f x 0 f x0.
: f [,] f(x) >0 x ,
: 0f x 0 0x ,
48. f -
,
: f (x) 0 , x .
: f (x) 0 , x .
: f (x) 0 , x .
: f (x) 0 , x
: f x .
49. f R
:
:
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:
:
:
50. f , :
: .
: .
: .
: .
: .
