diaforikos-logismos-mathimatika-glikeiou-papastmatiou-schooltime.gr_.pdf
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ΔΙΑΦΟΡΙΚΟΣ Λ ΟΓΙΣΜΟΣ ΣΥΝΟΠΤΙΚΗ ΘΕΩΡΕΙΑ – ΜΕΘΟΔΟΛΟΓΙΑ – ΛΥΜΕΝΑ ΠΑΡΑΔΕΙΓΜΑΤΑ Φροντιστήριο Μ.Ε. «ΑΙΧΜΗ» Κ.Καρτάλη 28 Βόλος Τηλ. 2421302598
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Transcript of diaforikos-logismos-mathimatika-glikeiou-papastmatiou-schooltime.gr_.pdf
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.. . 28 . 2421302598
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/ ..
3.1 .............................................................................................................. 5 ................................................................................................. 7
1. 0x .............................................................................................. 7 2. .............................................................................................................................. 12 3. ..................................................................................................................... 13 4. .............................................................................................................................. 14 5. ............................................................................................................................ 15
3.2 ............................................... 19 3.3 ................................................................................................................ 20 .............................................................................................. 21
1. ........................................................................................................................... 21 2. ................................................................................................. 22 3. ............................................................................................................. 24 4. ............................................................................................................................ 25 5. .................................................................................................... 27 6. ........................................................................................ 28
3.4 ........................................................................................................ 29 .............................................................................................. 30
1. .................................................................... 30 2. x x ................................................................................................... 31 3. ........................................................................ 31 4. ........................................................................................ 32 5. ....................................................................................................... 33 6. .................................................................................................................... 35 7. ............................................................................................................................ 36
3.5 ROLLE .............................................................................................................................. 40 .............................................................................................. 41
1. Rolle ........................................................................... 41 2. .............................................................................................................................. 43 3. ( ) 0f x = f .............................................................................. 44 4. ....................................................................................................................................... 46 5. f , . Rolle F ............................................................ 47 6. F ( ) ( )F a F = ........................................................................................ 53 7. ( ) 0f x = ............................................................................................................. 55 8. v ........................................................................................................................................ 56 9. . Bolzano . Rolle f ..................................................................... 58
-
/ ..
10. . Rolle F f ................................................................................... 59 11. ..................................................................................................................... 61
3.6 ............................................................................................................... 62 .............................................................................................. 63
1. ... .................................................................................................. 63 2. f .................................................................................................................. 64 3. . Bolzano .. ... ................................................................................................................. 67 4. f .................................................................................................................................................. 69 5. ... ............................................................................................................................ 73 6. .................................................................................................................................................... 78
3.7 ................................................................................ 81 - ............................................................................................. 82
1. ......................................................................... 82 2. ..................................................................................................... 84 3. .................................................................................... 89 4. ( ) ( ) ( ) ( )f x g x f x h x + = .......................................................................................... 93 5. ........................................................................................................................................ 95
3.8 ............................................................................................................ 97 - ............................................................................................. 98
1. ................................................................................................................................ 98 2. ........................................................................................................................... 101 3. .................................................................................................................. 102 4. x x x + = ...................................................................... 105 5. f ................................................................................................................. 107 6. ............................................................................................................... 110 7. ......................................................................................................................... 115
3.9 FERMAT ............................................................................. 117 - ........................................................................................... 119
1. Fermat ............................................................................................. 119 2. Fermat ............................................................................................... 120 3. f ...................................................................................................................... 124 4. .......................................................................................................... 124 5. f ............................................................................................... 127 6. ........................................................................... 132 7. .............................................................................................................................. 137 8. ............................................................................................................................................... 141
3.10 ..................................................................................................... 142
f f f
f
3
-
/ .. - ........................................................................................... 144
1. ................................................................................................................. 144 2. ......................................................................................................... 147 3. ....................................................................................... 150 4. ....................................................................................... 152 5. ......................................................................................................................... 156
3.11 ...................................................................................................................................... 159
- ........................................................................................... 160 1. .................................................................................... 160 2. ........................................................................................................................... 161 3. - ............................................................................................... 163
3.12 De L Hospital .................................................................................................................... 164
- ........................................................................................... 165
1. 00
............................................................................................................................................. 165 2.
............................................................................................................................................ 166 3. ..................................................................................................... 166
-
/ ..
3.1
f ))(,( 00 xfxA fC .
0
0
0
)()(lim
xxxfxf
xx
, fC , .
f 0x ,
0
0 )()(lim0 xx
xfxfxx
. f 0x )( 0xf . :
0
00
)()(lim)(
0 xxxfxf
xfxx
=
.
. h
xfhxfxf
h
)()(lim)( 00
00+
=
. ( ) ( ) ( )( )
0 00 1
0
lim1h
f x h f xf x
x h
=
f 0x , 0x
0
0
0
)()(lim
xxxfxf
xx
,
0
0
0
)()(lim
xxxfxf
xx
+
( ) ( )( )0 0 0y f x f x x x = f 0x :
5
-
/ .. ) +=
0
0 )()(lim0 xx
xfxfxx
( ) ) +=
0
0 )()(lim0 xx
xfxfxx
=
+ 0
0 )()(lim0 xx
xfxfxx
, ) =
0
0 )()(lim0 xx
xfxfxx
+=
+ 0
0 )()(lim0 xx
xfxfxx
, fC ))(,( 00 xfxA 0xx = . f 0x , .
: f 0x , , , 0x .
-
/ ..
1. 0x . f 0 fx D :
( ) ( ) ( )0
00
0
limx x
f x f xf x
x x
=
(1) ( ) ( ) ( )0 00 0limh
f x h f xf x
h+
= (2) ( ) ( ) ( )( )
0 00 01
0
lim , 01h
f x h f xf x x
x h
=
(3) f 0x . . f 0x , . . (1), (2) (3), . 1. f 0x : ( ) 2f x x= , 0 0x =
: , (1) 0x :
( ) ( )
( )
0
2 2 2 20
0 0 00
0 0 0
0 1lim lim lim lim0
lim lim lim 0 1 0
x x x x x
x x x
f x f x x x xx x x x x
x xx xx x
= = = =
= = = =
( ) ( ) ( )
0
00 lim 0
0xf x f
fx
= =
2. f 0x ( )
2
2
3 5 6, 1
2 3, 1
x x xf x
x x
+ = + >
0 1x = : , :
7
-
/ .. ( ) ( ) ( ) ( ) ( )
1 1
1 11 lim lim
1 1x xf x f f x f
fx x +
= =
: ( ) 21 3 1 5 1 6 3 5 6 4f = + = + = ( ( )1f
). ( ) ( ) ( )( )2 2
1 1 1 1
1 1 3 23 5 6 4 3 5 2lim lim lim lim 11 1 1 1x x x x
f x f x xx x x xx x x x + +
= = = =
( ) ( ) ( ) ( )( )
( )( )
( )( )( )
( )( )( )
( )( )( )( )
( )( )( )
2 2 22
21 1 1 1
22 2
2 2
2 2 21 1 1
2 21 1
2 3 2 2 3 2 3 21 2 3 4lim lim lim lim1 1 1 1 3 2
2 3 2 2 3 4 2 1lim lim lim
1 3 2 1 3 2 1 3 2
2 1 1 2 1lim lim
1 3 2
x x x x
x x x
x x
x x xf x f xx x x x x
x x x
x x x x x x
x x x
x x x
+ + + +
+ + +
+ +
+ + + + + = = = =
+ +
+ + = = = = + + + + + +
+ += =
+ + +( )( )
2
2 1 11
1 3 23 2
+= =
+ ++
f 0 1x = :
( ) ( ) ( ) ( ) ( ) ( )1 1
1 11 lim lim 1 1
1 1x xf x f f x f
f fx x +
= = =
3. f 0x ( ) 2 2 1f x x x= + , 0 2x =
: 2x = f . . : ( )
3 3, 21, 2
x xf x
x x >
= +
( )0 02 3x f x= = ( ) ( ) ( )
2 2 2
2 3 23 3 3lim lim lim 32 2 2x x x
f x f xxx x x+ + +
= = =
( ) ( ) ( )
2 2 2
2 21 3lim lim lim 12 2 2x x x
f x f xxx x x + +
= = =
( ) ( ) ( ) ( )2 2
2 2lim lim
2 2x xf x f f x f
x x +
f 0 2x =
-
/ ..
4. f 0 0x = x : ( ) 2 22x x x f x x x + , :
) ( )0 0f = ) ( )0
2x
df xdx
=
= : ) f 0 0x = : ( ) ( ) ( ) ( ) ( )
0 0 00 lim 0 lim lim
x x xf f x f f x f x
+ = = =
x :
0x > ( ) ( ) ( )
2 2 22 2 22 2
x f xx x x x xx x x f x x x x f x xx x x x
+ + +
: 0
0
lim 2 0
lim 0 1 0 0
x
x
x
xx xx
+
+
=
+ = + =
( )
0lim 0x
f x+
= 0x <
( ) ( ) ( )2 2 2
2 2 22 2x f xx x x x xx x x f x x x x f x x
x x x x
+ + +
: 0
0
lim 2 0
lim 0 1 0 0
x
x
x
xx xx
=
+ = + =
( )
0lim 0x
f x
= ( ) ( )
0lim 0 0 0x
f x f
= = ) ( ) ( ) ( )
0
0 0 2x
df xf f
dx=
= = : ( ) ( ) ( ) ( ) ( )
0 0
00 lim 0 lim
0x xf x f f x
f fx x
= =
2x :
( ) ( ) ( )2 2 2
2 22 2 2 2
22 2 1x f x f xx x x x x xx x x f x x x
x x x x x x
+ + +
0lim 2 2 1 2x
xx
= = 2
0lim 1 1 1 2x
xx
+ = + =
9
-
/ ..
( ) ( )0
lim 2 0 2x
f xf
x= = .
5. ( ) 2 4 2f x x x + x , f 0 ( )0 1f = : x 0x = . :
( ) ( ) ( ) ( )0
2 24 2 0 0 0 4 2 0 0 0 0x
f x x x f f f =
+ + = ( ) ( ) ( ) ( )
0 0
00 lim lim
0x xf x f f x
fx x
= =
:
( ) ( ) ( )( ) ( )
2 2 2
2 2
4 2 4 2 4 2
4 2 4 2
f x x x x f x x x
x x f x x x
+ + +
+ + + +
: 0x > :
( ) ( ) ( ) ( )
( ) ( )
22
2 2
22
4 2 4 24 2 4 2
4 2 4 2
x x f x x xx x f x x xx x x
x f xx x xx x x x x
+ + + + + + + +
+ + + +
: ( )( )
( ) ( )2 2
2 2
2 20 0 0
20
4 2 4 24 2lim lim lim4 2 4 2
lim 14 2
x x x
x
x xx x x x xx x x xx x x x
x xxx
+ + +
+
+ + + + + = + = + + + + +
= + = + +
( )( )( ) ( )
2 22 2
2 20 0 0
20
4 2 4 24 2lim lim lim4 2 4 2
lim 14 2
x x x
x
x xx x x x xx x x xx x x x
x xxx
+ + +
+
+ + + + + = + = + + + + +
= + = + +
( )
0lim 1x
f xx+
= 0x < :
-
/ ..
( ) ( ) ( ) ( )
( ) ( )
22
2 2
22
4 2 4 24 2 4 2
4 2 4 2
x x f x x xx x f x x xx x x
x f xx x xx x x x x
+ + + + + + + +
+ + + +
: ( )( )
( ) ( )2 2
2 2
2 20 0 0
20
4 2 4 24 2lim lim lim4 2 4 2
lim 14 2
x x x
x
x xx x x x xx x x xx x x x
x xxx
+ + + + + = + = + + + + +
= + = + +
( )( )( ) ( )
2 22 2
2 20 0 0
20
4 2 4 24 2lim lim lim4 2 4 2
lim 14 2
x x x
x
x xx x x x xx x x xx x x x
x xxx
+ + + + + = + = + + + + +
= + = + +
( )
0lim 1x
f xx
= (2) (1) (2) :
( ) ( ) ( )0 0
lim lim 1 0 1x x
f x f xf
x x + = = =
6. f, g 0 1x = ( ) ( )1 1f g= ( ) ( )2 1f x x g x+ + x , ( ) ( )1 2 1g f = + : f g 0 1x =
( ) ( ) ( ) ( ) ( )1 1
1 1lim lim 1
1 1x xf x f f x f
fx x+
= =
( ) ( ) ( ) ( ) ( )1 1
1 1lim lim 1
1 1x xg x g g x g
gx x+
=
. : ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( )( ) ( ) ( )
1 12 21 1 1 1 1 1 1 1
f g
f x x g x f x f x g x g f x f x x g x g=
+ + + + + :
1 0x > : ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( ) ( )
1 1 1 11 1 1 1
1 1 11 1
11 1
f x f x x g x gf x f x x g x g
x x xf x f g x g
xx x
+ + + +
+ +
( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1lim lim 1 lim 1 2 1
1 1x x xf x f g x g
x f gx x+ + +
+ + + (1)
1 0x < : 11
-
/ .. ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( ) ( )
1 1 1 11 1 1 1
1 1 11 1
11 1
f x f x x g x gf x f x x g x g
x x xf x f g x g
xx x
+ + + +
+ +
( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 1lim lim 1 lim 1 2 1
1 1x x xf x f g x g
x f gx x
+ + + (2)
(1) (2) : ( ) ( )1 2 1f g + = : x, ( )h x , . : ( ) 0h x > ( ) 0h x <
2. f , , 0x . f 0x :
f 0x ( ) ( ) ( )0 0
0lim limx x x xf x f x f x + = =
( ) ( ) ( ) ( ) ( )
0 0
0 00
0 0
lim limx x x x
f x f x f x f xf x
x x x x +
= =
, . 7. ( )
2
, 0
4, 0
a x xf x
x x x
+ = + + >
, f 0. : f 0. f 0 f 0. :
( ) ( ) ( )0 0
lim lim 0x x
f x f x f+
= = : ( )0 0f a a= + =
( ) ( )0 0
lim limx x
f x a x a
= + = ( ) ( )2
0 0lim lim 4 2x x
f x x x+ +
= + + = 2a = f 0 :
-
/ ..
( ) ( ) ( ) ( ) ( )0 0
0 0lim lim 0
0 0x xf x f f x f
fx x +
= =
:
( ) ( )0 0 0
0 2 2lim lim lim 10x x x
f x f x xx x x
+ = = =
(1)
( ) ( )
( )( )( ) ( )
( )
2 2
0 0 0
2 22
2 20 0
20
0 4 2 4 2lim lim lim0
4 2 4 2lim lim
4 2 4 2
lim 24 2
x x x
x x
x
f x f x x x xx x x x
x x x
x x x x
xx
+ + +
+ +
+
+ + + = = + =
+ + + = + = + = + + + +
= + = + +
(1) (2) : 1 =
3. f 0x : ( ) ( )
00limx x f x f x =
!!! f 0x 0x f 0x f 0x 8. f 1, ( ) ( ) ( )2 3 2g x x f x= + 1.
: f 1 ( ) ( )
1lim 1x
f x f
= 1x = : ( ) ( ) ( )21 1 3 2 1 0g f= + = : ( ) ( ) ( )
( ) ( ) ( )( )( )( )
( )
( )( )( ) ( )( )
( )( )( ) ( ) ( )
2 2 2
1 1 1 2
2
21 1 12 2
3 2 3 2 3 211 lim lim lim
1 1 1 3 2
1 1 11 1lim lim lim23 21 3 2 1 3 2
x x x
x x x
x f x x xg x gg f x
x x x x
x x fx xf x f x f xxx x x x
+ + + + = = = = + +
+ + = = = = + + + + + +
g 0 1x = ( ) ( )11 2fg =
13
-
/ .. 9. f 0 ( )
0lim 5x
f x xx
+= :
( )0 4f = : f 0 0x = : ( ) ( )0lim 0x f x f = ( )
0limx
f x
( ) ( ) ( ) ( )
f x xg x f x xg x x
x
+
= = ( )0
lim 5x
g x
= ( ) ( )( )
0 0lim lim 0x x
f x xg x x
= = : ( )0 3f =
( ) ( ) ( ) ( ) ( ) ( )0 0 0 0
00 lim lim lim lim 5 1 4
0x x x xf x f f x xg x x xf g x
x x x x
= = = = = =
4. . 10. f ( ) ( )1 1 2f f = = . : )
( )221
4lim
2xf xx x
+ +
) ( )1
2lim
3 2xxf x
x
+
: ) f 1 :
( ) ( ) ( ) ( )1 1
1 21 lim lim
1 1x xf x f f x
fx x
= =
. :
( ) ( )( ) ( )( )( )( )
( )( )( )
( )( )( )
( ) ( ) ( ) ( )
2
21 1 1
1 1
2 2 2 24lim lim lim
2 1 2 1 2
2 2 1 2 8lim lim 11 2 1 2 3
x x x
x x
f x f x f x f xf xx x x x x x
f x f x ff
x x
+ += = =
+ + + +
+ += = =
+ +
) :
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( )( )( )( )( )
( )( )( )( )( )
1 1 1
1 1
2 2 2lim lim lim
3 2 3 2 3 2 3 2
1 3 2 2 3 21 2lim lim
3 2 3 2 3 2 3 2 3 2 3 2
x x x
x x
xf x xf x f x f x xf x f x f xx x x x
f x x x f x xf x x f xx x x x x x
+ = = + =
+ + + + + + + + = + = + = + + + + + + + +
-
/ ..
( )( )( ) ( )( )( )
( )( ) ( )( ) ( )
( )( ) ( )( )
1
1
1 3 2 2 3 2lim
1 1
2lim 3 2 3 2
1
1 1 3 2 1 1 3 2 16
x
x
f x x x f x x
x x
f xf x x x
x
f f
+ + + + = + =
= + + + + + =
= + + + + + =
11. f 2 : ( ) ( )2
2
4 2lim
2xf x x f
x
: f 2 :
( ) ( ) ( )2
22 lim
2xf x f
fx
=
:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( )
( ) ( )( ) ( )( )( ) ( ) ( )( ) ( )( )( )
( ) ( ) ( )( ) ( ) ( ) ( ) ( )
2 2
2 2 2
2 2
2 2 2
4 2 2 2 24 2 4 4 2 4 2 2lim lim lim
2 2 24 2 4 22 2 2 2 2 2
lim lim2 2 2 2
2 2lim 4 2 2 4lim 2 lim 2 4
2 2
x x x
x x
x x x
f x f f x xf x x f f x f f x fx x xf x f f x ff x x f x x
x x x x
f x f f x ff x f x
x x
+ + + = = =
+ +
= + = =
= + = + = ( ) ( )2 2f f
12. :f ( )0 0f = . f 0, ( )1lim 0
xxf f
x+ =
: f 0 : ( ) ( ) ( ) ( )
0 0
00 lim lim
0x xf x f f x
fx x
= =
1 1y xx y
= = x + 0y : ( ) ( ) ( )
0 0
1 1lim lim lim 0x y y
f yxf f y f
x y y+ = = =
5.
f 0x : ( ) ( ) ( )
0
00
0
limx x
f x f xf x
x x
=
15
-
/ .. ( ) ( ) ( )0 00 0limh
f x h f xf x
h+
= ( ) ( ) ( )0 00 0limh f x h f xf x h + = 0 ( ) ( ) ( )( )
0 00 1
0
lim1h
f hx f xf x
x h
=
0 0x
. 13. f, g 0
( ) ( )2 2 22f x g x x+ = x , : ( ) ( )2 20 0 2f g + = . : x 0x = .
( ) ( ) ( ) ( ) ( )0
2 2 2 2 22 0 0 0 0 0x
f x g x x f g f=
+ = + = = ( )0 0g = f g 0 :
( ) ( ) ( ) ( )0 0
00 lim lim
0x xf x f f x
fx x
= =
( ) ( ) ( ) ( )0 0
00 lim lim
0x xg x g g x
gx x
= =
0x :
( ) ( ) ( ) ( ) ( ) ( )2 22 2
2 2 22 22 2 2
f x g x f x g xf x g x x
x x x x
+ = + = + =
( ) ( ) ( ) ( ) ( ) ( )
2 2 2 22 2
0 0 0 0lim lim 2 lim lim 2 0 0 2x x x x
f x g x f x g xf g
x x x x + = + = + =
14. f 1 ( )
0
1lim 5h
f hh+
= ,
( )1 0f = f 1. : f 1 : ( ) ( )
11 lim
xf f x
=
1 1x h h x= + = 0h 1x . : ( ) ( )
0 1
1lim 5 lim 5
1h xf h f x
h x +
= =
( ) ( ) ( ) ( )( )1
1f x
g x f x g x xx
= =
( )1
lim 5x
g x
= :
( ) ( )( ) ( )1 1
lim lim 1 1 0x x
f x g x x f
= = :
( ) ( ) ( ) ( ) ( )1 1
11 lim lim 5 1 5
1 1x xf x f f x
f fx x
= = = =
f 1 ( )1 5f =
-
/ ..
15. :f ( )1 0f = . ( )
( )( )
1 , 2
3 5 , 2
f x xg x
f x x
= >
2. : f 1 :
( ) ( ) ( ) ( ) ( )1 1
1 11 lim lim 0
1 1x xf x f f x f
fx x
= =
f 1 1 ( ) ( )
11 lim
xf f x
=
g 2 . : ( ) ( ) ( )2 2 1 1g f f= =
2x < ( ) ( ) ( ) ( )
2 2
2 1 1lim lim
2 2x xg x g f x f
x x
=
(1) 1 1y x x y= = + 2x : 1y . (1) : ( ) ( ) ( ) ( ) ( ) ( )
1 1
1 11 lim lim 1 0
1 2 1y yf y f f y f
fy y
= = = =
+
2x > ( ) ( ) ( ) ( )
2 2
2 3 5 1lim lim
2 2x xg x g f x f
x x+ +
=
(2)
53 53
yy x x += = 2x : 1y . (1) : ( ) ( ) ( ) ( ) ( ) ( )
1 1
1 12 lim lim 3 3 1 05 12
3y y
f y f f y ffy y
= = = =
+
( ) ( ) ( ) ( ) ( )
2 2
2 2lim 0 lim 2 0
2 2x xg x g g x g
gx x +
= = =
16. f 0 ,x y ( ) ( ) ( ) 5f x y f x f y xy+ = + + , f 0x .
: ,x y 0x y= = ( ) ( ) ( ) ( ) ( ) ( ) ( )
0
5 0 0 0 0 5 0 0 0x y
f x y f x f y xy f f f f= =
+ = + + + = + + = f 0 :
( ) ( ) ( ) ( )0 0
00 lim lim
0x xf x f f x
fx x
= =
0x .
( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0 0 0 00 0 0 05 5
lim lim limh h h
f x h f x f x f h x h f x f h x hf x
h h h + + + +
= = =
17
-
/ .. ( ) ( ) ( )0 0 00 0
5lim lim 5 0 5h h
f h f hx h x f xh h h
= + = + = +
f 0x ( ) ( )0 00 5f x f x = + 17. f 1 ( ), 0,x y + ( ) ( ) ( )f x y f x f y = + , f 0 0x > .
: ( ), 0,x y + 1x y= = ( ) ( ) ( ) ( ) ( ) ( ) ( )
1
1 1 1 1 1 0x y
f x y f x f y f f f f= =
= + = + = f 1 : f ( )0 0,x + .
( ) ( ) ( )( )( ) ( ) ( )
( )( )( )
( )( )
( )0 0 0 00 1 1 1 1
0 0 0 0 0
11lim lim lim lim1 1 1 1h h h h
f x h f x f x f h f x f h f h ff x
x h x h x h x h x +
= = = = =
f 0 0x > ( ) ( )0
0
1ff x
x
=
( ) ( ) ( ) ( )1 1
11 lim lim
1 1x xf x f f x
fx x
= =
-
/ ..
3.2 :
f . : H f , , ,
Ax 0 . f ),( , ),(0 x . f ],[ , ),(
( ) ( )limx
f x fx
+
( ) ( )lim
x
f x fx
.
( ) ,f x c c= ( ) 0f x = ( )f x x= ( ) 1f x = ( ) 1 , 0f x x
x= ( ) 21 , 0f x xx =
( ) , 0f x x x= > ( ) 1 , 02
f x xx
= > ( ) vf x x= ( ) 1vf x v x = ( )f x x= ( )f x x = ( )f x x= ( )f x x = ( ) ,
2f x x x k = + ( ) 21 ,f x x = ( ) 21 , 2f x x x k = + + ( ) ,f x x x k = ( ) 21 ,f x x = ( ) 21 ,f x x x k = ( ) xf x e= ( ) xf x e = ( ) xf x a= ( ) lnxf x a a = ( ) ln , 0f x x x= > ( ) 1 , 0f x xx = > ( ) ln , 0f x x x= ( ) 1 , 0f x x
x =
19
-
/ .. 3.3
gf , 0x , gf +
0x : )()()()( 000 xgxfxgf +=+
gf , 0x , gf 0x :
)()()()()()( 00000 xgxfxgxfxgf +=
gf , 0x 0)( 0 xg , gf 0x :
20
00000 )]([
)()()()()(
xgxgxfxgxf
xgf
=
g 0x f )( 0xg , gf 0x :
)())(()()( 000 xgxgfxgf =
( )( ) ( )a f x a f x = ( ) ( )( )21 f xf x f x = ( )( ) ( )
( )2f x
f xf x
= ( )( ) ( ) ( )1v vf x v f x f x =
( )( ) ( ) ( )f x f x f x = ( )( ) ( ) ( )f x f x f x = ( )( ) ( )( )2
f xf x
f x
= ( )( ) ( )( )2f xf x f x =
( )( ) ( ) ( ) lnf x f xf x a = ( )( ) ( ) ( )f x f xe f x e = ( )( ) ( )( ) ( )ln , 0
f xf x f x x
f x = ( )( ) ( )( ) ( )ln , 0f xf x f x xf x = >
-
/ ..
1. f . ( ) v af x x= 0x > : ( ) ( ) 1
a a a vv va a vv v va a af x x x x x x
v v v
= = = = =
!!! . 1. ) ( )
3 2
3 2x xf x ax = + ) ( ) lnf x x x=
) ( ) 2 lnf x x x x= ) ( ) 2ln1x xf x x= + : ) f . f . :
( ) ( ) ( ) ( ) ( ) ( )3 2 3 2
3 2
2 2
1 13 2 3 2 3 2
1 13 2 13 2
x x x xf x ax ax x x a x
x x a x x a
= + = + = =
= =
) f ( )0,fD = + f . f ( )0,fD = + ( )0,fD = + . :
( ) ( ) ( ) ( ) 1ln ln ln 1 ln ln 1f x x x x x x x x x xx
= = + = + = + ( )0,x + ) f ( )0,fD = + f . f ( )0,fD = + ( )0,fD = + .
( ) ( ) ( ) ( ) ( )2 2 2 22 2 2
ln ln ln ln
12 ln ln 2 ln ln
f x x x x x x x x x x x x x
x x x x x x x x x x x x x x x xx
= = + + =
= + + = + +
21
-
/ .. ) f ( )0,fD = + f . f ( )0,fD = + ( )0,fD = + .
( )( ) ( ) ( )( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( )( )
( )( )( )
( )
( )
2 2
22 2
2 2
22
22 2
2 22 2
2 2
22
ln 1 ln 1ln1 1
ln ln 1 ln 1
1
11 ln 1 ln 2 0 ln 1 1 2 ln
1 1
ln ln 1
1
x x x x x xx xf xx x
x x x x x x x x
x
x x x x x x x x x xxx x
x x x x
x
+ + = = = + +
+ + + = =+
+ + + + + = = =+ +
+ + +=
+
( )( )
2 2
22
ln ln 1
1
x x x xf xx
+ + + =+
2.
. . . ( ) ( )( ) ( )g xh x f x= ( ) ( ) ( )lng x f xh x e= . . 2. ) ( ) ( )325 1f x x= + ) ( )
24
21 ln
f xx
=+
) ( ) ( ) 2 12 xf x x e += + ) ( ) ( )2 2ln 1f x x= + ) ( ) ( )ln1 xxf x e= + : ) f . f . :
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 3 1 2 2 22 2 2 2 2 2 25 1 3 5 1 5 1 3 5 1 5 1 3 5 1 10 30 5 1f x x x x x x x x x x = + = + + = + + = + = +
-
/ ..
) f 0x > 2 21 ln 0 ln 1x x+ . ( )0,fD = + f ( )0,+ ( )0,+ : ( ) ( ) ( )
12 4
24
2 2 1 ln1 ln
f x f x xx
= = +
+
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )
1 1 512 2 2 2 24 4 4
5 52 24 4
524
1 12 1 ln 2 1 ln 1 ln 1 ln 1 ln4 2
1 1 2ln ln1 ln 2ln ln 1 ln2 2 1 ln
f x x x x x x
x xx x x xx x
= + = + + = + + =
= + = + = +
) f f . :
( ) ( )( ) ( )( ) ( ) ( ) ( )2 2 2 21 1 1 2 12 2 2 2 1 2 2 2x x x xf x x e x e x x e x x xe + + + + = + = + = + + = + ) 2 21 0 1x x+ > > x . f . f . :
( ) ( )( ) ( )( )( ) ( ) ( )( ) ( ) ( )
( ) ( )
222 2 2 2 2 2
2
22
2 2
1ln 1 ln 1 2ln 1 ln 1 2ln 1
14 ln 122ln 1
1 1
xf x x x x x x
xx xxx
x x
+ = + = + = + + = + =+
+= + =
+ +
) f 0x > 1 0xe + > . ( )0,fD = + f ( )0,+ ( )0,+ . : ( ) ( ) ( ) ( )ln ln ln 11
xx x exf x e f x e += + = ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
ln ln 1 ln ln 1 ln ln 1
lnln ln 1 ln ln 1
ln ln 1 ln ln 1 ln ln 1
1 ln 1 ln 11 ln lnln 1 ln 11 1 1
x x x
x x
x e x e x ex x x
x x xx xxx e x ex xx x x
f x e e x e e x e x e
e e ex e x ee e x e ex e x e x e
+ + +
+ +
= = + = + + + = + + + = + + = + = + + + + +
23
-
/ .. 3.
:f A . : 1:f A 1A A , x ( )f x 1:f A 2 1A A , ( ) ( )( )f x f x = - ( ) :v vf A , 1v vA A A :
( ) ( ) ( ) ( )( )1v vf x f x = 3v f fD , . 3. ( ) ( )2 lnf x x x x = + + :
f ( )0,fD = + f ( )0,fD = + ( )0,fD = + . :
( ) ( )( ) ( ) ( ) ( )( ) ( )
( ) ( )
2 2 1ln ln 2
12
f x x x x x x x x xx
f x x xx
= + + = + + = + +
= + +
f ( )0,fD = + ( )0,fD = + . :
( ) ( ) ( ) ( )( ) ( )
( ) ( )
22
22
1 1 12 2 2
12
f x x x x x xx x x
f x xx
= + + = + + =
=
4. ( )P x , : ) ( ) ( ) 2 2P x P x x x = , x ) *v ( ) ( )2P x P x = x ( )1 0P = : ) ( )P x ( )P x 1 . 2 2,
( )P x 2. ( ) 2P x ax x = + +
-
/ ..
( ) 2P x ax = + : ( ) ( ) ( )2 2 2 2 22 2 2 2 2
1 12 1 1
2 1
P x P x x x ax x ax x x ax a x x x
a
= + + = + + =
= = = = = =
( ) 2 1P x x x= + ) : ( ) ( )2 2 1P x v = ( )P x v= : ( )2 1 2 2 2v v v v v = = =
( ) 2P x ax x = + + ( ) 2P x ax = + ( )1 0 0P a = + + = (1) :
( ) ( ) ( )( )( )
2 2 2 2 2 2 2
2
2 2
2 4 4
4 1 044 4 1 0
P x P x ax ax x a x a x ax x
a aa
= + = + + + + = + + = =
= = = =
=0 ==0 ( )P x .
14
a = : 2 2
2 22
11 1 144 4 410 04
1 1 10 040 2 2
a
== = = = = = = = + + = + = = + + =
( )
2 14 2 4x xP x = +
4. 5. f 1 ( ) ( )3 21f x f x x+ + = x , ( )1 2f = . : 1x =
( ) ( ) ( ) ( )( ) ( )3 21 1 1 1 1 1 1 0 1 0f f f f f+ + = + = = ( )2 1 1 0f + = . :
25
-
/ .. ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )3 2 3 2 21 1 1 1 1f x f x x f x f x x f x f x x x+ + = + = + = +
1x : ( ) ( )( ) ( )( ) ( ) ( )
22
11 1 11 1
f x xf x f x x xx f x
++ = + =
+(1)
f 1 1, : ( ) ( )1
lim 1x
f x f
= (1) :
( )( ) ( ) ( ) ( )2 21 1
1 1 1lim lim 1 1 21 1 1 1x x
f x x f fx f x f
+ + = = = + +
6. :f
( ) ( ) ( )3 2 2f x xf x f x x + = x . ( )0f . : 0x = :
( ) ( ) ( ) ( ) ( ) ( ) ( )0
3 2 3 22 0 0 0 0 2 0 0 0x
f x xf x f x x f f f f =
+ = + = = f . :
( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( )
3 2 3 2
3 2
2 2 2
2 2
2 2
2 2
2 2
2 2
3 2 2
3 2 2 2
3 2 2 2
3 2
f x xf x f x x f x xf x f x x
f x xf x f x x x
f x f x x f x x f x f x x
f x f x f x xf x f x f x x
f x f x f x xf x f x x f x x
f x f x f x xf x f x
+ = + =
+ =
+ =
+ =
+ =
+ + ( ) 2 2f x x=
0x = : ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
02 2
02 2
3 2 2 2
3 0 0 0 2 0 0 0 0 2 2 0
0 2
x
x
f x f x f x xf x f x f x x
f f f f f f
f
=
=
+ + =
+ + =
=
7. :f . : ) f , f . ) f , f . :
) f x : ( ) ( )f x f x = :
-
/ ..
( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )
f x f x f x f x f x x f x f x f x
f x f x
= = = =
=
f . ) f x : ( ) ( )f x f x = : ( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )
( ) ( )f x f x f x f x f x x f x f x f x
f x f x
= = = =
=
f . : .
5. , x y , :
y x, x y.
8. :f , ( ) ( ) ( )y xf x y e f x e f y a+ = + + ,x y
: ) ( )0 0f a= = ) ( ) ( ) ( )0 xf x f x f e = + x : ) 0x y= = : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
00 00 0 0 0 0 2 0 0
x yy xf x y e f x e f y a f e f e f a f f a f a
= =
+ = + + + = + + = + = y x= : ( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( ) ( )
00
0
0 0
0 0 1 0 0
yy x x
f ax x
f x y e f x e f y a f x e f x e f a
f x f x e f f e f
=
+ = + + + = + +
= + =
x , ( )0 0f = ) x y. ( ) ( ) ( ) ( )( ) ( ) ( )( )
( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( )
y x y x
y x y x
y x
f x y e f x e f y a f x y e f x e f y a
f x y x y e f x e f y a f x y f x e e f y
f x y e f x e f y
+ = + + + = + +
+ + = + + + = +
+ = +
0y = : ( ) ( ) ( )0xf x f x e f = +
27
-
/ .. 6.
f ( )1 :f f . 1f ( )0x f ( )( )1 0 0f f x . 1f : ( )( )1f f x x = ( )x f , : ( )( ) ( ) ( )( )( ) ( ) ( ) ( )
( )( )1 1 1 1
1
11f f x x f f x f x f xf f x
= = =
(1) ( )0 0f x y= ( )0 0f x , : ( ) ( ) ( )
10
0
1f yf x
=
9. :f 1f . ( )
2
11
f xx
=+
1f , ( ) ( ) ( )1 1f x f x = x . : 1 :f , ( )( )1f f x x = , :
( )( ) ( ) ( )( )( ) ( ) ( ) ( )( )( )
( ) ( ) ( )( )
1 1 1 11
21 1
11
1
f f x x f f x f x f xf f x
f x f x
= = =
= +
(1) ( )1f , . : ( ) ( ) ( )( ) ( ) ( )
( )( )( )( )( )
( ) ( ) ( ) ( )( )( )( )
( ) ( )( ) ( )( )
( )( )( ) ( ) ( )
21 1 121 1 1 1
2 21 1
21 1
1 1 1
21
1 21
2 1 2 1
2 1
2 1
f x f x f xf x f x f x f x
f x f x
f x f xf x f x f x
f x
+ = + = = + +
+ = =
+
-
/ ..
3.4
( ) ( )( )0 0 0y f x f x x x = f f f ( )f x = f 0x : ) +=
0
0 )()(lim0 xx
xfxfxx
( ) ) +=
0
0 )()(lim0 xx
xfxfxx
=
+ 0
0 )()(lim0 xx
xfxfxx
, ) =
0
0 )()(lim0 xx
xfxfxx
+=
+ 0
0 )()(lim0 xx
xfxfxx
, fC ))(,( 00 xfxA 0xx = .
29
-
/ ..
1. f (x, y) . f. f(x)=y.
( ) ( ) ( )AAA xxxfxfy = f . x=xA Cf . Cf M(x0, y0). (): ( ) ( ) ( )000 xxxfxfy = (x, y) (). ( ) ( )( )0A00A xxxfxfy = (1) (1) x0.
1. f(x)=x2 3x+5. x0 (2,0). : f(2)=22 32+5=30. (2,0) Cf. M(x0, y0) f . (): ( ) ( ) ( )000 xxxfxfy = (2,0) :
( )( ) ( ) ( )( )
=
+=
=
+=
=++=+
=+=
32x
32x
2124x
2124x
01x4xx36x2x45x3x
x23x25x3xx2xf)x(f0
0
0
0
0
0200
2000
20
00020000
2. ( ) 24f x x= .
( ) 1:16
y = . : f
( ) 8f x x = x ( )0 0,M x y ( ) 1: 16y = . 0 116y = 0x =
1,16
M
.
-
/ ..
f. ( )( )1 1,A x f x f. fC ( )( )1 1,A x f x ( ) ( ) ( )( ) ( )2 21 1 1 1 1 1 1 1: 4 8 8 4y f x f x x x y x x x x y x x x = = = ()
2 21 1 1 1
1 18 4 4 8 016 16
x x x x = = (1) (1) 1x ( )2 218 4 4 64 1
16 = = +
2
1,18 64 1
8x + += 21,2 8 64 18x +=
( ) ( )1,1 1,2 1f x f x =
( ) ( ) ( )( )( )
2 22 2
1,1 1,2
22 2 2 2
8 64 1 8 64 18 8 8 64 1 8 64 18 8
8 64 1 64 64 1 1
f x f x
+ + + = = + + + =
= + = =
( ) 1:
16y = .
2. x x f. f xx. f(x0)=0 x0. ( )0xfy = 3. ( ) xlnxxf = ( )+ ,0x . xx. :
f ( ) ( ) 1xlnxlnxxf +== xx ( ) 0xf 0 =
: ( ) 11xln0 ex1xln01xln0xf ee ====+= ( ) 11 eyefy ==
3. f ():y=x+.
31
-
/ .. Cf
(). ( ) = 0xf ().
Cf ().
( ) 1xf 0 = (). 4. f(x)=x2 4x +3 : ) f(x) ) i) y=6x+5 ii) 3x
21y =
:
) ( ) ( ) 4x23x4xxf 2 =+= ) CIF y=6x+5 ( ) 5x10x264x26xf 0000 ====
( ) 835455f 2 =+=
( ) ( )( ) ( ) 22x6y5x68y5x5f5fy === Cf 3x
21y =
( ) ( ) 1x2x224x22xf121xf 00000 =====
( ) ( ) ( ) 831411f 2 =+=
( ) ( )( ) ( ) 6x2y1x28y1x1f1fy +=+=+= 4.
f Cf y=x+
( )( ) ( )
0
0 0 0
f x
f x x f x
=
=
( )( )
0
0 0
f x a
f x ax
=
= +
5. ( ) +++= xxxxf 23 : ) f(x) ) f (1, f(1)) ) , y=2x+1 f . :
) ( ) ( ) 1x2x3xxxxf 223 ++=+++= ) f
-
/ ..
( ) +== 241f ) ( ) ++=+++= 21111f 23 y=2x+1 f :
( )( ) ( )
=
=
=++
=+
=
=
61
1422242
11f11f21f
f ( ) 6xxxxf 23 ++= 6. ( ) xf x a= , 0a > . ( ) : 0x y = fC . : f . ( ) lnxf x a a = ( )( )0 0,M x f x f. ( ) : y x = fC ( )( )0 0,M x f x
( )( )
0
0 0
f x a
f x ax
=
= + .
0 0 0
00 0 0
0 0 0
00 0 0
0 01
ln 1ln 1 ln 1ln ln
1ln
x x x
x ex x x
e
x a x ea a a a e x ee a ea x a ea x a x a x
x e x e
a a ee
= = = = = = == == = =
= = = =
1ea e=
5. f(x) g(x).
. Cf (, f()) Cg B(, g()).
( ) ( )( ) ( ) ( ) ( )
==
ggffgf
(, ) : ( ) ( )( ) ( ) ( ) ( )
=
=
ggffgf
(, ) f(x)=g(x). 7.
( ) 2xxf = ( )x1xg =
:
33
-
/ .. : ( ) x2xf = ( ) 2x1xg = Cf Cg (, f()) B(, g()) : ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )2 xggy:
1 xffy:
2
1
=
=
( ) ( )( ) ( ) ( ) ( )
==
ggffgf .
: ( )
=
=
=
=
=
=
=
=
=
=
212
81
21
24
12
1
22
1
112
12
3
2
4
2
2
2
22
2 (1) (2) :
( ) ( )( ) ( ) 4x4y2x44y2x2f2fy +=== 8. ( ) 2x5x3xf 2 += ( )
x1xxg = .
, . :
M(x, y) Cf Cg. 2x5x3y 2 += x
1xy = . :
01xx5x31xx2x5x3x
1x2x5x3 23232 =++=+=+ (1) (1) x=1 0111513 23 =++ . (1) Horner x=1. (1) : ( )( ) ( )( )
31- x 1x0
31x1x1x01x2x31x 2 ===
+=
1(1,0)
4,
31M 2
( ) 5x6xf 2 = ( ) ( ) ( ) ( ) 22 x1x x1xx1xx 1xxg == = 75
316
31f =
=
9
31g =
Cf
Cg .
4,
31M 2
3 -5 1 1 x=1 3 -2 -1 3 -2 -1 0
-
/ ..
( ) ( )1g11f == Cf Cg. 1(1,0) : ( ) ( )( ) 1xy1x1f1fy ==
6. f.
f . ( ) = 0xf x0.
f xx . ( ) = 0xf x0.
9. ( )x
1xxf = . : ) f(x) ) Cf xx 450. ) Cf
41
. :
) : ( ) ( ) ( )( ) 222 x1
x1xx
xx1xx1x
x1xxf =+=
=
=
) Cf xx 450 ( ) ( ) 1x1x1
x145xf 0
202
0
00 ====
: x=1 : ( ) ( ) ( )( ) ( ) 1xy1x10y1x1f1fy:1 === x= 1 : ( ) ( ) ( )( ) ( ) 3xy1x12y1x1f1fy:2 +=+=+= ) Cf
41
. ( ) 2x4x
41
x1
41xf 1
212
11 ====
: x=2 : ( ) ( ) ( )( ) ( ) x
41y2x
41
21y2x2f2fy:3 ===
x= 2 : ( ) ( ) ( )( ) ( ) 2x41y2x
41
23y2x2f2fy:4 +=+=+=
35
-
/ .. 7.
10. :f ( )0 0f > :
( )2 2 14
f xx = x . ( ) : 2 2y x = +
fC . :
( )2 2 14
f xx = x .
f. :
( ) ( ) ( ) ( )2
2 2 2 21 4 1 2 14
f xx f x x f x x = = + = +
f ( ) 22 1f x x= + ( ) 22 1f x x= + ( )0 0f > . ( ) 22 1f x x= + x f
( ) ( ) ( ) ( ) ( )2
2
2 2
1 22 1 22 1 1
x xf x x f x f xx x
+ = + = =
+ +
() fC : ( )( ) ( ) ( )
( ) ( )( )
( )( ) ( ) ( ) ( )
2 222
2
2 2 2 2
2 22 2 2 12 2 11
2 2 2 2 2 22 2
4 2 1 12 1
2 2 22 22 2
xf x x xx x
xf x x f x x f x xf x x
x x xx xf x xf x xf x x
= = = += + + = + = + = += +
= + = = + = += + = +
1x = (2) ( ) 21 2 1 2 2 1 1 2 2 2 2 2 2f = + + = =
1x = ( ) ( ) ( )21 2 1 2 2 1 1 0 2 2 0f = + + = =
1x = 1x = () fC 11. :g ( ) 0g x x ( ) ( )( )g xf x g x= , x . x x , fC fC x x .
:
-
/ ..
f . :
( ) ( )( ) ( )( )( ) ( ) ( )
( )( )
2
2
g x g x g xg xf x f x
g x g x
= =
fC x x ( ) ( ) ( )( )
( )( )
0 00
0 0 00
,0 0 0 0g x
f
g xA x A C f x g x
g x
= = =
0x x= ( )f x :
( )( )( ) ( ) ( )
( )( )( )
( )( )( )( )
( )2 2
0 0 0 00 0 02 2
0 0
1g x g x g x g x
f x f x f xg x g x
= = =
x x , fC ( )0 ,0A x : ( )0 1 4f x
= = = 12. ( ): 0,f + ( )2 3f x x= 0x > . fC 0 2x = . : f.
02
xy x x y
>
= = f : ( ) ( ) ( ) ( ) ( )
23 3
2 3y x
f x x f y y f x x=
= = = ( )0,x + f ( )0,+ ( )0,+ .
( ) ( ) ( )3 3 12 23 3
2 2xf x x f x x f x
= = =
fC 0 2x = : ( ) ( ) ( )( ): 2 2 2y f f x = : ( ) ( ) ( )32 2 2 2 2f f= = ( ) 3 22
2f =
( ) ( ) ( )( ) ( )3 2 3 2: 2 2 2 2 2 2 22 2
y f f x y x y x = = =
37
-
/ .. 13. ( ) : 2 1y x = + fC 0 1x =
( )21
1lim
1xf x
x
+.
: () fC 0 1x = :
( )1 2f = ( ) ( ) ( )1 2 1 1 1 1f f = + = :
( ) ( ) ( ) ( )1 1
1 11 2 lim 2 lim 2
1 1x xf x f f x
fx x +
= = =+ +
:
( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( )( )
2
1 1 1 1 1
1 1 1 11lim lim lim 1 lim lim 1
1 1 1 12 1 1 4
x x x x x
f x f x f x f xf xf x f x
x x x xf
+ + += = = =
+ + + += = 14. ) 8 8xe x= (1, 2). ) ( ) 22f x x= ( ) xg x e= . : ) ( ) 8 8xh x e x= + [ ]1, 2x
h [1, 2] [1,2]. ( )1 0h e= > , ( ) 22 8 0h e= < ( ) ( )1 2 0h h <
Bolzano ( )0 1, 2x ( )0 0h x = 8 8xe x= (1, 2). ) f g . ( )( )1 1,A x f x f. fC . ( ) ( ) ( )( ) ( )2 21 1 1 1 1 1 1 1 1: 2 4 4 2y f x f x x x y x x x x y x x x = = = ( )( )2 2,B x g x g. gC B . ( ) ( ) ( )( ) ( ) ( )2 2 2 2 2 2 22 2 2 2 2 2 2: 1x x x x x x xy g x g x x x y e e x x y e x x e e y e x e x = = = + = + ( )1 ( )2 :
( ) ( ) ( ) ( )
( ) ( )( )
2 2 2 2 1
2
222
01 1 1 1
2 221 1 21 1 2 1 1 21 2
221
1 2 1 21 2
4 4 4 42 2 2 02 4 1 2 4 1 02 1
8 8 14 2 242 2 2 2 22 2
x x x x x
x
xxx
x e x e x e x ex x xx x x x x xx e x
e xx ex ex x x xx x
= = = = + == = =
= == = = =
(1) 2 0x x= (1, 2)
-
/ ..
2 0
1 02 2x xx x=
= ( ) ( )1 2 0 0, 2 2,x x x x=
f g ( ) ( ) ( )( ) ( )0 00 0 0 0: 1x xy g x g x x x y e x e x = = +
39
-
/ .. 3.5 ROLLE
(Rolle) f : ],[ ),( )()( ff = , , ),( , :
0)( = f , , , ),( , fC ))(,( fM x.
y
O x
(,f ())
(,f ()) (,f ())
18
-
/ ..
1. ROLLE
. Rolle f [, ] : f [, ] f (, ) ( ) ( )f a f =
( ),a , ( ) 0f = . ) ( ) ( )f a f = ) , Rolle . . f ( ),a ( ) 0f = [, ] Rolle. 1. ( ) 2
4 1, 18 20 9, 1x x
f xx x x
-
/ .. Rolle ( ),a ( ) 0f = , ( )0,2 fC x x 2. f [0, ], (0, ) ( ) ( )0 0f f + =
) ( ) ( )g x f x x= , [ ]0,x Rolle. ) ( )0, , ( ) ( )f f = : ) :
g [0, ] [0, ]. g (0, )
(0, ). ( ) ( )( ) ( ) ( ) ( )g x f x x g x f x x f x x = =
( ) ( ) ( )( ) ( ) ( )0 0 0 0g f f
g f f
= =
= = ( ) ( ) ( ) ( )0 0 0f f f f + = =
( ) ( )0g g = Rolle [0, ]. ) ( )0, ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 0g f f f f
f f f f
= = =
= =
3. f, ( ) ( )f x xf x= x . :
) ( ) ( )g x f x= Rolle [0, ]. ) ( )0, ( ) ( )f g = : ) Rolle g [0, ]
g [0, ] [0 , ] g (0, )
(0, ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )g x f x g x f x x g x x f x = = =
( ) ( ) ( )0 0 0g f f= = , ( ) ( ) ( )0g f f = = ( ) ( )0g g = ) Rolle ( ) ( )0, : 0g =
-
/ ..
( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )
( ) ( ) ( )g x f x
f x xf x f x x f x x f x f x xg x g x
= = = + = +
x = : ( ) ( ) ( )
( )( ) ( )
0g
f g g f g
=
= + = 2.
f , [, ] Rolle, :
( ) ( )f a f = f [, ] 0x ,
f . f (, ),
. 4. , ,
( ) 24 2 , 1
4 1, 1ax x
f xx x x
+ =
+ >
Rolle [ ]3,3 :
( ) ( ) ( ) ( )23 3 4 3 2 3 4 3 1 12 2 10 12 6 6 5 1f f a a = + = + + = + + = f
f [ ) ( ]3,1 1,3 . f 0 1x =
( ) ( ) ( ) ( ) ( ) ( )0 0
01 1
lim lim lim lim 1x x x x x x
f x f x f x f x f x f + +
= = = = :
( ) ( )1 1
lim lim 4 2 4 2x x
f x ax a
= + = + ( ) ( )2
1 1lim lim 4 1 2 4x x
f x x x + +
= + = ( )1 4 2f a = + ( )4 2 2 4 2 2 1 2a a + = + + =
f f ( ) ( )3,1 1,3 f 0 1x =
( ) ( ) ( ) ( ) ( )1 1
1 1lim lim 1
1 1x xf x f f x f
fx x +
= =
:
43
-
/ .. ( ) ( ) ( )
1 1 1 1
1
1 4 14 2 4 2 4 4lim lim lim lim1 1 1 1
lim 4 4x x x x
x
f x f a xax a ax ax x x x
a a
+ = = = =
= =
( ) ( ) ( )( ) ( )
( )( )
2 2
1 1 1 1
1
1 1 1 4 14 1 2 4 1 4 4lim lim lim lim1 1 1 1
1 1 4lim 2 4
1
x x x x
x
f x f x x xx x x xx x x x
x xx
+ + + +
+
+ + + += = = =
+
= =
4 2 4 2 2 1a a = + = (3) (2) (3) :
2 2 12 2 1 1 1 0
aa
+ =
+ + = + = = (1) :
6 6 5 6 6 5 + + = + = (4) (2) (4) :
2 2 1 6 6 36 6 5 6 6 5a a + = + =
+ = + =
8 212 812 3
= = =
16
= 3. ( ) 0f x = f
Rolle f (, ).
f [, ] ( ),a x x . 5. ( ) ( ) ( )1 ln 1f x x x= + . : ) ( ) 0f x = (0, 1) ) ( ) 1 11 x xx e+ + = (0, 1) : ) f ( )1,fD = +
f [0, 1] [0, 1] f (0, 1)
(0, 1) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( )1 ln 1 1 ln 1 1 ln 1f x x x f x x x x x = + = + + +
( ) ( ) ( ) ( ) ( ) ( )1 1ln 1 1 ln 11 1
x xf x x x f x xx x
+ = + + = + ++ +
-
/ ..
( ) ( ) ( )0 0 1 ln 0 1 0f = + = , ( ) ( ) ( )1 1 1 ln 1 1 0f = + = ( ) ( )0 1f f= Rolle ( )0,1 , ( ) 0f = ) x = :
( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( )( )1 1 1
1 1 1ln 1 ln 1 0 ln 1 1 ln 1 11 1 1
ln 1 1 1
f
e
+ +
= + + + + = + = + + = + + +
+ = + =
( )0,1 ( ) 1 11 x xx e+ + = (0, 1). 6. ( ) ( )
32
3 2axf x x x = + + + +
,
, , , 03 2 + + = . ( )0,1
f ( )( ), f x x . : f .
f [0, 1] f (0, 1)
( ) ( ) ( ) ( )3
2 2 23 2 2
axf x x x f x ax x = + + + + = + + +
( )0f = , ( ) 03 213 2 3 2
a
a af
+ + =
= + + + + = + + + = ( ) ( )0 1f f= Rolle ( )0,1 , ( ) 0f = , ( )0,1 f ( )( ), f x x . 7. 3 2 0x ax x + + + = , 2 3a > . : 1 2 3, ,x x x 1 2 3x x x< < ( ) 3 2f x x ax x = + + + [ ]1 3,x x x
f [ ]1 2,x x [ ]2 3,x x . f [ ]1 2,x x [ ]2 3,x x .
( ) 23 2f x x ax = + + ( ) ( ) ( )1 2 3 0f x f x f x= = =
45
-
/ .. Rolle ( ) ( )1 1 2 1, : 0x x f = ( ) ( )2 2 3 2, : 0x x f = 1 2 < f :
( )2 20 2 4 3 0 3a a > > > 8. f 1-1, , ( ) 0f = : f 1-1
1 2,x x 1 2x x ( ) ( )1 2f x f x= . 1 2x x<
f [ ]1 2,x x f ( )1 2,x x ( ) ( )1 2f x f x=
Rolle ( )1 2,x x ( ) 0f =
4. , ( ) :
( )c x = ( ) ( ) ( )( )f x xf x xf x + = 1
1
vv axax
v
+ = +
( ) ( ) ( ) ( ) ( ) ( )( )f x g x f x g x f x g x + = 2
a ax x
=
( ) ( ) ( )2xf x f x f xx x = ( )1 ln x
x= ( ) ( ) ( ) ( )
( )( )( )2
f x g x f x g x f xg x g x
=
( )x x = ( ) ( ) ( )2
2f x
f x f x
=
( )x x = ( ) ( ) ( )1
1
vv f xf x f x
v
+ =
+
( )x xe e = ( ) ( ) ( )( )f x f xe f x e =
-
/ ..
( )21 x
x
= ( )( ) ( )( )lnf x f xf x =
( )21 x
x
= ( ) ( ) ( ) ( ) ( )( )2f x f x f x f x f x + =
axax ee
a+
=
( ) ( ) ( )( )
( )( )
2
2
f x f x f x f xf xf x
=
( ) ( )x
ax
+ + =
( ) ( ) ( )
( )( )( )
2
2
f x f x f x f xf xf x
=
( ) ( )ax
ax
+ + =
( ) ( ) ( ) ( ) ( ) ( )g x g xf x g x f x f x e e + =
ln1 xx
+ = +
( ) ( ) ( ) ( ) ( ) ( )g x g xf x g x f x f x e e = ln
xx aa
a
=
. . , , .
5. f , . ROLLE F Bolzano, Rolle ,
( ),a , . x . , ( )g x ( )G x ( ) ( ) ( ), ,G x g x x a = .
g. ( )G x Rolle,
. 9. 6 7 3 ln 2a = + , ( )2x a x
x = +
(1, 2) : ( ) ( )2f x x a x
x = ( )1,2x
f, F f ( ) ( )f x F x= .
47
-
/ .. :
2 1 32 2
2 1 3x xx x
+ = = +
( ) ( )
xx
=
1 ln xx=
f ( ) ( ) ( ) ( )3 3ln ln3 3
xx xF x a x F x a x x
= = +
[ ]1,2x
F [1, 2] [1, 2] F (1, 2)
(1, 2). ( ) ( ) ( ) ( )( )( )
( ) ( )
3 2
2
3ln3 3x xF x a x x F x a x x
x
F x x a xx
= + = +
=
( ) 11
3F a= , ( ) 82 ln 2
3F a = +
( ) ( )1 2F F= ( ) ( ) 1 81 2 ln 2 1 3 8 3 3 ln 2 6 7 3 ln 2
3 3F F a a a = = + = + = . Rolle F , ( )1,2 ( ) ( )0 0F f = =
( )2x a xx = + (1, 2)
10. f [1, 2] (1, 2) ( ) ( )2 1 ln 2f f = . ( )1,2 ( )
22 3 1f + =
: x . : ( ) ( ) ( )2 22 3 1 2 3 1 10 2 3 0x x x xf x f x f x x
x x x + + = = + =
. ( ) ( ) 12 3g x f x x
x= + ( )1,2x G g
( ) ( )G x g x = ( )1,2x . :
-
/ ..
( ) ( )( )f x f x = , ( )22x x = , ( )3 3x = , ( )1 ln xx = ( ) ( ) 2 3 lnG x f x x x x= + [ ]1, 2x
G [1, 2] [1, 2] G (1, 2)
(1, 2). ( ) ( )( ) ( ) ( )2 13 ln 2 3G x f x x x x G x f x x x
= + = + ( )1,2x ( ) ( )1 1 2G f= + , ( ) ( )2 2 2 ln 2G f= +
( ) ( )1 2G G= ( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 2 2 ln 2 2 1 ln 2G G f f f f= + = + = .
Rolle G, ( )1,2 : ( ) ( ) ( )
22 3 10 0G g f + = = =
11. f 10,2
( )0 0f = .
10,2
( ) ( ) ( )1 2 2f f = : x . : ( ) ( ) ( ) ( ) ( ) ( )1 2 2 1 2 2 0x f x f x x f x f x = = (1) ( )1 2 2x = (1) : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 2 0 1 2 1 2 0 1 2 0x f x f x x f x x f x x f x = + = = ( ) ( ) ( ) ( )1 2 2g x x f x f x= 10,
2x
( ) ( ) ( )1 2G x x f x= g.
G 10,2
10,2
G 10,
2
10,2
( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 1 2 2G x x f x G x x f x f x = =
( ) ( ) ( ) ( )0 1 2 0 0 0 0G f f= = = , 1 1 11 2 02 2 2
G f = =
( ) 102
G G =
Rolle G, 10,
2
:
49
-
/ .. ( ) ( ) ( ) ( ) ( )0 0 1 2 2G g f f = = =
12. f [, ], (, ) ( ) 0f x ( ),x a , ( ),a ( )
( )1 1f
f a
= +
.
: x . :
( )( )
1 1f xf x a x x
= +
x a= x =
1a x
1x
.
. ( )( )
( )( ) ( )( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )2
1 1 2
2 0
f x f x a x x a x x f x a x f xf x a x x f x a x x
a x x x f x a x f x
+ = + = = +
+ + =
( ) ( ) ( ) ( ) ( )2 2g x a x x x f x a x f x = + + ( ),x a : ( ) ( )2 2 2a x x x a x a x + = + = + g : ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )
2
2 2
2
2g x a x x x f x a x f x
g x a x x x f x a x x x f x
g x a x x x f x
= + +
= + + +
= +
G g ( ) ( ) ( ) ( ) ( )( ) ( )2G x a x x x f x G x a x x f x = + =
[ ],x a G [, ] [, ] G (, )
(, ) ( ) ( )( ) ( ) 0G a a a a f a= = , ( ) ( )( ) ( ) 0G a f = = ( ) ( )G a G =
Rolle G, ( ),a : ( ) ( ) ( )( )
1 10 0f
G gf a
= = = +
13. f [0, 1], (0, 1)
( ) ( )1 1 10 1f f = . ( ) 0f x [ ]0,1x
( )0 0,1x ( ) ( )20 0 02f x x f x = . :
-
/ ..
x 0x . :
( ) ( ) ( )( )( ) ( )
( )0
22 22 2 2 0
f xf x f xf x xf x x x
f x f x
= = =
( ) ( )( )22
f xg x x
f x
= ( )0,1x :
( )( ) ( )2
1f xf x f x
=
( )22x x = g ( ) ( ) 21G x xf x= [ ]0,1x
G [ ]0,1 [0, 1] G (0, 1)
(0, 1) ( ) ( ) ( )
( )( )
22
1 2f x
G x x G x xf x f x
= =
( ) ( )100
Gf
= , ( ) ( )11 11G f= ( ) ( ) ( ) ( ) ( ) ( )
1 1 1 10 1 1 10 1 0 1
G Gf f f f
= = = . Rolle G, ( )0 0,1x :
( ) ( ) ( ) ( )20 0 0 0 00 0 2G x g x f x x f x = = = 14. f Rolle [1, 4]. * : ( ) ( )21 1 3 2
3 2f f
+ =
: x :
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
22 2
2
1 1 3 22 2 3 3 3 2 2 2 3 3 3 2 0
3 23 3 2 2 2 3 0
f x f xx xf x f x x xf x f x
xf x x f x x
+ = = + =
+ + =
( ) ( ) ( )23 3 2 2 2 3g x f x x f x x = + + ( )1,4x :
( )( ) ( ) ( ) ( )3 2 3 2 3 2 3 3 2f x f x x f x = = ( )( ) ( ) ( ) ( )2 2 2 22f x f x x x f x = =
( )2 3 2 3x x x + = +
51
-
/ .. g ( ) ( ) ( )2 23 2 3G x f x f x x x= + + [ ]1, 4x f . Rolle
f [1,4] f (1, 4) ( ) ( )1 4f f=
( ) ( ) ( )2 23 2 3G x f x f x x x= + + [ ]1, 4x : G [1, 4] [1, 4] G (1, 4)
[1, 4] ( ) ( ) ( )( ) ( ) ( ) ( )2 2 23 2 3 3 3 2 2 2 3G x f x f x x x G x f x x f x x = + + = + +
( ) ( ) ( ) ( )2 21 3 1 2 1 1 3 1 2 1 2G f f f= + + = + , ( ) ( ) ( ) ( )
( ) ( )( ) ( )
1 42 22 3 2 2 2 2 3 2 2 4 2 2 1 2 1
f f
G f f f f G=
= + + = + = + = Rolle G, ( )1,4 :
( ) ( )( ) ( )21 1 3 20 0
3 2f f
G g
+ = = =
15. :f ( ) ( )1f x ef x+ = x . ( )0,1 ( ) ( )f f = .
: x .
( ) ( ) ( ) ( ) 0f x f x f x f x = = xe :
( ) ( ) 0x xe f x e f x = ( ) ( ) ( )x xg x e f x e f x = ( )0,1x : ( )x xe e = g : ( ) ( ) ( ) ( ) ( )( )x x xg x e f x e f x e f x = + = g ( ) ( )xG x e f x= [ ]0,1x
G [0, 1] [0, 1] G (0, 1)
(0, 1). ( ) ( )11 1G e f= , ( ) ( )0 0G f=
1x = ( ) ( )1f x ef x+ = ( ) ( ) ( ) ( ) ( ) ( )11 0 1 0 0 1f ef e f f G G= = =
Rolle G, ( )0,1 : ( ) ( ) ( ) ( )0 0G g f f = = =
-
/ ..
16. f [0, 1], (0, 1) ( ) 0f x > [ ]0,1x . ( )0 ln 2f = ( ) ( )1 ln 1f e= +
( )0,1 ( ) ( )( )2 1 ff e = . : x = .
( ) ( )( ) ( ) ( ) ( )( )( ) ( )
( )
( ) ( )( )2 1 2 1 2 2 01 1
f x f xf x f x f x
f x f x
e f x e f xf x x e e f x x e x x
e e = = = =
( ) 1f xe ( ) ( ) ( )00 1 0f x f xf x e e e> > > [ ]0,1x
( )( ) ( )
( ) 21
f x
f x
e f xg x x
e
=
( )0,1x :
( )( ) ( ) ( )1f x f xe e f x = , g ( ) ( )( )( ) ( ) ( )( )( )21 2 ln 11f x f xf xeg x x g x e xe = = g ( ) ( )( ) 2ln 1f xG x e x= [ ]0,1x
G [0, 1] [0, 1] G (0, 1)
(0, 1). ( ) ( )( )( ) ( )
( ) ( )( )
2ln 1 21
f xf x
f x
e f xG x e x G x x
e = =
( ) ( )( ) ( )0 ln 20 ln 1 0 ln 1 0fG e e= = = ,
( ) ( )( ) ( )( ) ( )1 ln 11 ln 1 1 ln 1 1 ln 1 1 1 ln 1 0f eG e e e e+= = = + = = ( ) ( )0 1G G= Rolle G, ( )0,1 :
( ) ( ) ( ) ( )( )0 0 2 1 fG g f e = = = 6. F ( ) ( )F a F =
, . . a x= Rolle . 17. f ( ) ( )f a fa = 1 a < < . ( ),a ( ) ( ) lnf f = : :
53
-
/ .. ( ) ( ) ( ) ( ) ( ) ( )ln ln ln lnf a f f a fa a f a a f = = =
( ) ( ) lnF x f x x= [ ],x a F [, ] [, ] F (, )
(, ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ln ln ln lnf xF x f x x F x f x x f x x F x f x x
x = = + = +
( ) ( ) ( ) ( )ln lnF a f a a f F = = = Rolle , F ( ), :
( ) ( ) ( ) ( ) ( )0 ln 0 lnf
F f f f
= + = = 18. f [, ] ( ) 0f x [ ],x a . ( ) ( ) ( ) ( )f a f f f a = ( ),a ( ) ( ) 0f f > : , ( ) ( )f a f ( ) 0f x [ ],x a . : ( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( )( ) ( )
( )( )
( )( )
f a f f f a f a ff a f f f a
f a f f a f f a f
= = =
( ) ( )( )f x
F xf x
=
[ ],x a F [, ] [, ] F (, )
(, ) ( ) ( )( ) ( )
( ) ( ) ( )( )( )( )
2
2
f x f x f xf xF x F x
f x f x
= =
( ) ( )( )
( )( ) ( )
f a fF a F
f a f
= = =
Rolle , F ( ), :
( )( ) ( ) ( )( )
( )( )( ) ( ) ( )( ) ( ) ( ) ( )( )
( ) ( )
22 2
20 0 0
0
f f fF f f f f f f
f
f f
= = = =
>
-
/ ..
7. ( ) 0f x = f [, ]. f ( ) ( ) ( )f a f f = = [, ] Rolle [, ] [, ] Rolle ( )1 ,a ( )2 ,
( ) ( )1 2 0f f = = Rolle [ ]1 2, f ( ) ( )1 2, ,a ( ) 0f = ( ) 0f x = . 19. f ( ) ( )1 1 1f f = = ( )0 0f = . ( )1,1 fC ( )( ),A f ( ) : 2 3y x = : ( )1,1 ( ) 2f = x = ( ) ( )( )2 0 2 0f x f x x = = ( ) ( ) 2g x f x x= [ ]1,1x g ( ) ( ) 2G x f x x= [ ]1,1x ( ) ( ) ( )2G x f x x g x = =
G [-1, 0] [0, 1] [-1,0] [0, 1]
G (-1, 0) (0, 1) (-1, 0) (1, 0)
( ) ( ) ( ) ( )21 1 1 1 1 0G f f = = = , ( ) ( )0 0 0G f= = , ( ) ( ) 21 1 1 0G f= = ( ) ( ) ( )1 0 1G G G = =
Rolle ( ) ( ) ( )1 1 11,0 : 0 0x G x g x = = ( ) ( ) ( )2 2 20,1 : 0 0x G x g x = =
g : g [ ]1 2,x x [ ]1 2,x x g ( )1 2,x x
( )1 2,x x ( ) ( )1 20g x g x= =
Rolle ( ) ( )1 2, 1,1x x ( ) ( )0 2g f = = ( )1,1
fC ( )( ),A f ( ) : 2 3y x = .
55
-
/ .. 8. v
:
: ( ) 0f x = ( ) 0f x = +1 :
1 2 1... +< < < < :
1. [ ] [ ]1 2 1, ,..., , + Rolle, f :
( ) ( )1 1 2 1, ,..., , + 2. f ( ) 0f x =
, . , :
3. f [ ] [ ]1 2 1, ,..., , f 1 ( f ), ( )3f 2
f ( )vf , ( )1vf . .
20. f, ( ) : 2 1 0x y + = . fC 2y x= . : fC ( ) : 2 1 0 2 1x y y x + = = + ( ) 2f x x . fC 2y x= ( ) 2f x x= .
1 2,x x 1 2x x< ( )1 12f x x= ( )2 22f x x=
( ) ( ) 2g x f x x= [ ]1 2,x x x g [ ]1 2,x x [ ]1 2,x x g ( )1 2,x x
( )1 2,x x ( ) ( )( ) ( ) ( )2 2g x f x x g x f x = =
( ) ( )1 20g x g x= = Rolle ( )1 2,x x ( ) ( )0 2g f = =
-
/ ..
fC 2y x= . 21. f
( ) ( ) ( )2 3f x xf x e f x x e = + x . ( ) 0f x = . : ( ) 0f x = 1 2,x x 1 2x x< ( )1 0f x = ( )2 0f x = f [ ]1 2,x x f ( )1 2,x x ( ) ( )1 20f x f x= =
Rolle ( )1 2,x x ( ) 0f = :
( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )2 3 22 3f x f xx xf x e f x x e f x f x f x e f x x e = + = + x x = :
( ) ( ) ( ) ( ) ( ) 2 22 3 3 0ff f f e f e e = + + = 23 0e + > ( ) 0f x = . 22. f ( ) 2f x x . ( ) 2f x x x = + , . :
( ) ( )2 2 0f x x x f x x x = + + = ( ) ( ) 2g x f x x x = + x ( ) 0g x = 1 2 3, ,x x x 1 2 3x x x< < ( ) ( ) ( )1 2 3 0g x g x g x= = =
g [ ]1 2,x x [ ]2 3,x x [ ]1 2,x x [ ]2 3,x x
g ( )1 2,x x ( )2 3,x x ( )1 2,x x ( )2 3,x x ( ) ( ) 2g x f x x =
( ) ( ) ( )1 2 3 0g x g x g x= = = Rolle :
( ) ( )1 1 2 1, : 0x x g = ( ) ( )2 2 3 2, : 0x x g = g [ ]1 2, [ ]1 2,
57
-
/ .. g ( )1 2,
( )1 2, ( ) ( ) 2g x f x =
( ) ( )1 20g g = = Rolle ( ) ( ) ( )1 2, : 0 2g f = = ( ) 2f x x . ( ) 2f x x x = + , .
9. . BOLZANO . ROLLE f ( ) 0f x = , f [, ] (, ), :
Bolzano [, ]. ( ) ( ) 0f a f < ( )0 ,x a ( )0 0f x =
( )1 ,x a 1 0x x ( )1 0f x = . Rolle f [, ], ( ) ( )0 1, : 0x x f = . , ( ) 0f x = (, )
(, ) Bolzano, (, ) Rolle ( ) 0f x = (, )
23. f [0, 1] ( )1 2f x< < [ ]0,1x ( ) 2f x x ( )0,1x , ( )0 0,1x
( ) 20 0 1f x x= + . : ( ) ( ) 2 1g x f x x= [ ]0,1x
g [0, 1] [0, 1] ( ) ( )0 0 1g f= : ( )1 2f x< < [ ]0,1x 0x =
( ) ( ) ( )1 0 2 0 0 1 1 0 0f f g< < < < > ( ) ( )0 1 2g f= : ( )1 2f x< < [ ]0,1x 1x =
( ) ( ) ( )1 1 2 1 1 2 0 1 0f f g< < < < < ( ) ( )0 1 0g g <
Bolzano ( )0 0,1x ( ) ( ) 20 0 00 1g x f x x= = + .
( )1 0,1x ( )1 0g x = 0 1x x< g [ ]1 2,x x [ ]1 2,x x
-
/ ..
g ( )1 2,x x ( )1 2,x x ( ) ( )( ) ( ) ( )2 1 2g x f x x g x f x x = =
( ) ( )0 10g x g x= = Rolle ( ) ( )0 1, 0,1x x ( ) ( )0 2g f = = ( ) 2f x x ( )0,1x .
( )0 0,1x ( ) 20 0 1f x x= + 10. . ROLLE F f
f [, ] ( ) 0f x = [, ] , Rolle.
F f Rolle F [, ]. ( ) 0f x = 0x (, )
( ) 0f x = (, ) 1x 0 1x x< . ( ) ( )0 1 0f x f x= = . Rolle f ( ) ( ) ( )0 1, , : 0x x a f = . .
24. f ( ) ( )0 1f f e = ( ) xf x e x . ( )0,1 ( ) 1f e = + . : x = :
( ) ( )1 1 0x xf x e f x e = + = ( ) ( ) 1xg x f x e= [ ]0,1x g ( ) ( ) xG x f x e x= ( ) ( )G x g x = :
G [0, 1] [0, 1] G (0, 1)
(0, 1). ( ) ( )0 0 1G f= , ( ) ( )1 1 1G f e= .
( ) ( ) ( ) ( ) ( ) ( )0 1 0 1 1 1 0 1G G f f e f f e= = = . . Rolle ( )0 0,1x
( ) ( )0 00 0G x g x = = ( ) ( )1 10,1 : 0x g x = 0 1x x< ( )
59
-
/ .. g [ ]0 1,x x [ ]0 1,x x g ( )0 1,x x
( )0 1,x x . ( ) ( ) xg x f x e =
( ) ( )0 1 0g x g x= = . Rolle ( )0 1,x x : ( ) ( ) ( )0 0g f e f e = = = ( ) xf x e
x . ( ) ( )0,1 : 1f e = + 25. f, [0, 1] ( ) 2f x [ ]0,1x . f Rolle [0, 1],
( )0,1 fC ( ) 2g x x x= .
: . f Rolle, :
f [0,1] f (0,1) ( ) ( )0 1f f=
( )0,1 fC ( ) 2g x x x=
( ) ( ) ( ) 2 1f g f = = ( ) ( ) 2 1h x f x x= + [ ]0,1x h ( ) ( ) 2H x f x x x= + ( ) ( )H x h x = [ ]0,1x
( )H x [0, 1] [0, 1]. ( )H x (0, 1)
(0, 1). ( ) ( )0 0H f= , ( ) ( )1 1H f= ( ) ( )0 1H H=
Rolle ( ) ( ) ( )0,1 : 0 0H h = = ( ) ( )1 10,1 : 0x h x = 1x <
h [ ]1, x [ ]1, x h ( )1, x
( )1, x . ( ) ( )( ) ( ) ( )2 1 2h x f x x h x f x = + =
-
/ ..
( ) ( )1 0h h x = = . Rolle
( ) ( ) ( ) ( )0 0 0 00,1 : 0 2 0 2x h x f x f x = = = , ( ) 2f x [ ]0,1x . ( )0,1 ( ) 1f e = + .
11. 26. f, g ( ) ( ) ( ) ( )f x g x f x g x x . 1 2,x x g ( )1 2x x< , :
) 1 2,x x f. ) ( )3 1 2,x x x f. : ) 1 2,x x g ( ) ( )1 20g x g x= = Bolzano g ( )1 2,x x . 1 2,x x f ( ) ( )1 20f x f x= = ( ) ( ) ( ) ( )f x g x f x g x x 1x x= : ( ) ( ) ( ) ( )1 1 1 1 0 0f x g x f x g x . 1x f.
2x f. ) f [ ]1 2,x x ( ) 0f x [ ]1 2,x x x : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0f x g x f x g x f x g x f x g x
( ) ( ) ( ) ( )( )
( )( )2
0 0f x g x f x g x g x
f x f x
( ) ( )( )
g xh x
f x= [ ]1 2,x x x
h [ ]1 2,x x [ ]1 2,x x h ( )1 2,x x
( )1 2,x x ( ) ( )( )
11
1
0g x
h xf x
= = , ( ) ( )( )22 2 0g xh x f x= = ( ) ( )1 2h x h x= . Rolle ( ) ( ) ( ) ( ) ( ) ( )1 2, : 0 0x x h f g g f = = ( ) ( ) ( ) ( )f x g x f x g x x . ( )3 1 2,x x x f.
61
-
/ .. 3.6
( ...) f : ],[ ),( , , ),( , :
fff
=)()()(
, , , ),( ,
f ))(,( fM .
(,f ())
a x
y
M(,f ())
A(a,f (a))
20
-
/ ..
1. ...
. (...) [, ], : f [, ] (, )
, ( ),a , : ( ) ( ) ( )
f f af
a
=
... :
( ) ( ) ( ) ( )f f a a f = . , ... ( )( ),M f fC ,
( ),a fC ( )( ),A a f a ( )( ),B f . 1. f [, ], (, ) ( ) 0f x > [ ],x . : ) ( ) ( )lng x f x= [, ]. ) ( ),a ( )
( )( ) ( )( )
fa
ff ef
= :
) g [, ] ( ) 0f x > [ ],x g [, ] [, ] g (, )
[, ]. ( ) ( )( ) ( ) ( )( )ln
f xg x f x g x
f x = = ( ),x a
) : ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
ln ln, : ln
fa
f
g g a f f f a f fg a
a f a f f a
fe
f a
= = =
=
2. f [2, 3], ( )2,3 ( ) ( )3 2 2 3f f= . ( )2,3 ( ) ( )2 2f f = :
63
-
/ .. :
f [2, 3] f (2, 3)
... ( ) ( ) ( ) ( )3 2
2,3 :3 2
f ff
=
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )33 2 2 2 2 22
f f f f f f f f = = = 2. f
[, ]
[ ] [ ] [ ]1 1 2 1, , , ,..., ,a ... : . :
( ) ( ) ( ) ( )1 1 2 2 ... v vf f f f + + + = :
1 2 ... 0 = + + + d = [, ] 1 2, ,...,d d d
1 2 1, ,..., : 1 1d = , 2 1 2d = ,, 1v vd =
: 1 1d = + , ( )2 1 2 d = + + , , ( )1 1 2 1... d = + + + +
1 2 1, ,..., : 11 2 1
1 2
... d
= = = = , :
1 1d = + , ( )2 1 2 1 2d d d = + + = + + , , ( )1 1 2 1 1 2 1... ...d d d d = + + + + = + + + +
: ( )1 2 ... d = + + + + 1 1
... 1 2 1, ,..., , :
1 1d = , 2 1 2d = ,, 1v vd = ... . ( ) ( ) ( )1 2 ... vf f f + + + = , 1 2 ... 1 = = = = :
d
= , d = + 1 1 1 2 1, ,..., :
-
/ ..
1 2 1 1... d = = = = , d = . ( ) ( )f f < :
( ) ( ) ( )1 2
1 2
... vvf f f
+ + + =
( ) ( ),f a f 1 2, ,..., ( )1 2 1, ,..., ,
( ) ( ) ( ) ( )1 12 11 2
...f a f f f a
= = = = , 1 2 ... 0 = + + + ( )1 2 1, ,..., , , ( ) ,f a d 1 2, ,..., .
... [ ]1 2 1, ,..., ,vx x x a : ( )1 1f x = , ( )2 2f x = , . . . , ( )1 1f x =
... : { }1, x , { }1 2,x x , { }2 3,x x , . . . , { }1,vx .
2v = 3v = , , . 3. f ( ) ( )3 3f x f x= x . ( )1 3f = ( )1 2 3, , 0,3x x x ( ) ( ) ( )1 2 3 9f x f x f x + + = .
: ( ) ( ) ( )1 2 3 9f x f x f x + + = . B , . 1
3ad d = = [ ] [ ] [ ]0,1 , 1, 2 , 2,3
0x = ( ) ( ) ( ) ( )3 0 3 0 2 0 0 0 0f f f f = = = 1x = ( ) ( ) ( ) ( )3 1 3 1 3 3 3 3 9f f f f = = =
f [ ] [ ] [ ]0,1 , 1,2 , 2,3 f ( ) ( ) ( )0,1 , 1,2 , 2,3
... : ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1
1 00,1 : 1 0
1 0f f
x f x f x f f
= =
(1) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2
2 11,2 : 2 1
2 1f f
x f x f x f f
= =
(2) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 3 3
3 22,3 : 3 2
3 2f f
x f x f x f f
= =
(3) (1), (2) (3) :
65
-
/ .. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 2 3 1 0 2 1 3 2 3 0 9f x f x f x f f f f f f f f + + = + + = =
4. [ ]: , 2006f a a + . f ( ), 2006a a + ( ) ( )2006 2006f a f a+ = + , ( )1 2, , 2006a a + ( ) ( )1 2 2f f + = :
2006 2006 10032 2
a ad + = = = [ ] [ ], 1003 , 1003, 2006a a a a+ + +
f [ ] [ ], 1003 , 1003, 2006a a a a+ + + f ( ) ( ), 1003 , 1003, 2006a a a a+ + +
... : ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1
1003 1003, 1003 :
1003 1003f a f a f a f a
a a f fa a
+ +
+ = =+
(1) ( ) ( ) ( ) ( )( ) ( )1 1
2006 10031003, 2006 :
2006 1003f a f a
a a fa a
+ +
+ + =+ +
( ) ( ) ( )1
2006 10031003
f a f af
+ + = (2)
(1) (2) : ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )1 2
1003 2006 10031003 1003
2006 20062
1003 1003
f a f a f a f af f
f a f a f a f a
+ + +
+ = + =
+ + = = =
5. f Rolle [, ]. ( )1 2 3, , ,a
( ) ( ) ( )1 2 32 3 0f f f + + = : f Rolle [, ]. :
f [, ] f (, ) ( ) ( )f a f =
1 1 = , 2 2 = 3 3 = . 1 2 3 1 2 3 6 = + + = + + =
6ad d
= = 1 1 1 1
56 6
a ad += + = + = ( ) ( )2 1 2 2 2 21 2 6 2 2
a a ad += + + = + + = + =
-
/ ..
[ ] [ ] [ ]1 1 2 25 5, , , , , , , , , ,
6 6 2 2a a + + + +
f 5 5, , , , ,
6 6 2 2a + + + +
f 5 5, , , , ,
6 6 2 2a + + + +
... :
( )( )
( )( )
1 1 1
5 55 6 6, : 56
6 6
a af f a f f aa f fa aa
+ + + = = +
( )( )
1
566
af f af
a
+ =
(1)
( ) ( )2 2 2
5 55 2 6 2 6, : 5 2 26 2
2 6 6
a a a af f f fa f fa a a
+ + + + + + = = + + ( )2
52 63
a af ff
a
+ + =
(2)
( )( )
( )( )
3 3 32 2, : 2
22
a af f f fa f fa a
+ + + = = + (3)
(1) , (2) (3) : ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( )1 2 3
5 56 6 6 6 6 66 2 6 22 3
6 60
f f a
a a a af f a f f f ff f f
a a af f
a
=
+ + + + + + = + + =
= =
3. . BOLZANO .. ...
6. f [0, 1] (0, 1) ( )0 1f = ( )1 3f = . : ) ( )0 0,1x ( )0 2f x = ) ( )1 2, 0,1 ( ) ( )1 21 1 1f f + = :
67
-
/ .. ) f [0, 1] ( ) ( )0 1f f . ( ) ( )0 1 2 3 1f f= < < = . : ( )0 0,1x ( )0 2f x = .
( ) ( ) 2g x f x= [ ]0,1x . Bolzano. ) [0, 1] [ ]00, x [ ]0 ,1x 0x [0, 1] f. :
f [ ]00, x [ ]0 ,1x f ( )00, x ( )0 ,1x
... : ( ) ( ) ( ) ( ) ( ) ( )
01 0 1 1 0
0 0 1
0 2 1 10, :0
f x fx f f x
x x f
= = =
(1)
( ) ( ) ( ) ( ) ( ) ( )0
2 0 2 2 00 0 2
1 3 2 1,1 : 11 1
f f xx f f x
x x f
= = =
(2)
(1) (2) : ( ) ( ) 0 01 21 1 1 1x x
f f + = + =
( )1 2, 0,1 ( ) ( )1 21 1 1f f + = 7. [ ]: ,f a ( ) 0f x
( ),x a ( ) ( )f a f . : ) ( )0 ,x a ( ) ( ) ( )03 2f x f a f = + ) ( )1 2, ,a ( )( ) ( )( )1 0 2 02f x a f x = ) ( )1 2 3, , ,a ( ) ( ) ( )1 2 3
2 1 3f f f
+ =
: ) ( ) ( ) ( ) ( )3 2g x f x f a f = [ ],x a
g [, ] [, ]. ( ) ( ) ( ) ( ) ( ) ( )( )3 2 2g a f a f a f f a f = =
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )3 2g f f a f f f f f = = = ( ) ( ) ( ) ( )( )22 0g a g f a f = <
Bolzano ( ) ( )0 0, : 0x a g x = ( ) ( ) ( )03 2f x f a f = +
-
/ ..
) ... [ ]0,a x [ ]0 ,x
f [ ]0,a x [ ]0 ,x f ( )0,a x ( )0 ,x
... : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )
01 0 1 0 1 0
0
0 1 0 1
,
221
3 3
f x f aa x f x a f f x f a
x a
f a ff a fx a f f a x a f
= =
+ = =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
02 0 2 0 2 0
0
0 2 0 2
,
22
3 3
f f xx f x f f f x
xf a f f f a
x f f x f
= =
+ = =
(1) (2) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
0 1
0 1 0 2
0 1
3 22
3
f f ax f
x a f x ff a f
x a f
= =
=
) (1) (2) : ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )( )
( ) ( ) ( )( )
( ) ( ) ( )
( ) ( ) ( )
( )( )( ) ( )
( )( )
( ) ( )( ) ( ) ( ) ( ) ( ) ( )
020 2
0 2
00 1 1
10
0
2 0 0
2 10
1
1333
232 33 2
311 2 3 3
32
f f a xf f a fx f x f f f ax af a f f a f
x a f f f f a fx a
xf f f a a x x
f f f f a f f ax af f a f
= = = = = =
= + = + =
( ) ( )( )
( ) ( )2 131 2 a
f f f f a
+ =
(3) f [, ] f (, )
... ( ) ( ) ( ) ( ) ( ) ( ) ( )3 3 31, :
f f a aa fa f f f a
= =
(4)
(3) (4) : ( ) ( ) ( )1 2 32 1 3f f f + = 4. f
69
-
/ .. f [ ,] (, ). . 0x (, ) : ( ) ( ) ( )0f a f x f = = Rolle [ ]0,a x [ ]0 ,x . Rolle : ( ) ( )1 0 1, : 0a x f = ( ) ( )2 0 2, : 0x f = . ( ) ( )1 20f f = = . Rolle [ ]1 2, ( ) ( )1 2, : 0f = . 0x (, ) : ( ) ( ) ( ) ( )0 0
0 0
f x f a f f xx a x
=
... [ ]0,a x [ ]0 ,x ( ) ( ) ( ) ( )01 0 1
0
, :f x f a
a x fx a
=
( ) ( ) ( ) ( )02 0 20
, :f f x
x fx
=
( ) ( )1 2f f = . Rolle [ ]1 2, ( ) ( )1 2, : 0f = 8. f [1, 5] ( ) ( ) ( )1 5 2 3f f f+ = . ( )1,5 , ( ) 0f =
: [1, 5] 3 . ( ) ( ) ( )1 3 5f f f= = ... :
f [1, 3] [3, 5] f (1, 3) (3, 5)
... : ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 1
3 1 3 11,3 :
3 1 2f f f f
f f
= =
( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2
5 3 5 33,5 :
5 3 2f f f f
f f
= =
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )2 1
1 5 2 3 1 5 3 3 5 3 3 1
5 3 3 12 2
f f f f f f f f f f f
f f f ff f
+ = + = + =
= =
f [ ]1 2, f ( )1 2, ( ) ( )2 1f f =
-
/ ..
. Rolle ( ) ( )1 2, ,a , ( ) 0f = 9. f . fC ( )( ),A a f a fC ( )( ),B f , > : ) f 1-1 ) ( ),a ( ) 0f = : ) fC ( )( ),A a f a ( ) ( ) ( )( ): y f a f a x a = ( )( ),B f (). :
( ) ( ) ( ) ( )( ) ( ) ( ) ( )f f a
B f f a f a a f aa
= =
f [, ] f (, )
... : ( ) ( ) ( ) ( )0 0, :
f fx a f x
a
=
f 1-1 ( ) ( )0 0a x f a f x = ) f :
f [ ]0,a x f ( )0,a x ( ) ( )0f x f a =
. Rolle ( ) ( )0, : 0a x f = 10. f ( ) 0f x > x . , , ( < )
( ) ( ) ( ), ,f a f f , ( ),a
( ) ( ) ( )2f f f = : , , ,
2 += = + , 2 = + .
( ) ( ) ( ), ,f a f f ( ) ( ) ( ) ( )( )
( )( )
2 f ff f a ff a f
= = x = :
71
-
/ .. ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )( )
( )( )
22
