Θεματα μαθηματικα κατευθυνσησ γ λυκειου
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Transcript of Θεματα μαθηματικα κατευθυνσησ γ λυκειου
-
0
: . . 0
2012
: ,
-
1
: . . 1
1. , :f g \ \ ( ) 5( ) 1 ( ) (1)f g x x x g x= + + + x\ . ) g 1-1.
) 0x \ , ( )0 0f x x= . ) f g ,
g xx.
) ( ) ( )1 2 1 2g x g x x x= = ( ) ( ) ( ) ( ) (1)1 2 1 25 5
1 1 1 2 2 25 5
1 1 2 2
( ) ( )
1 ( ) 1 ( )
1 1 (2)
g x g x f g x f g x
x x g x x x g xx x x x
= = + + + = + + +
+ + = + +
: 5( ) 1x x x = + + 4( ) 5 1 0 ( ) ( ) "1 1"x x x x = + > /
"1 1"
1 2 1 2(2) ( ) ( )x x x x = =
) ( )x
[ ] ( ).( 1) (1) 0( 1) 1
1,1 : ( ) 01,1(1) 3
-
2
: . . 2
2. , :f g \ \ : (1) (2) ... (2006) (1) (2) ... (2006) (1)f f f g g g+ + + = + + + . [ ]0 1, 2006x , : 0 0( ) ( )f x g x= .
( ) ( ) ( ) ( )1 (1) (1) (2) (2) ... (2006) (2006) 0 (2)f g f g f g + + + = : ( ) ( ) ( )h x f x g x= [ ]1,2006 , - [ ]( ) , 1, 2006m h x M x
[ ][ ]
[ ]
( ) (2)
1 1, 2006 (1)
2 1, 2006 (2)2006 (1) (2) ... (2006) 2006
...2006 1, 2006 (2006)
m h M
m h Mm h h h M
m h M
+
+ + +
2006 0 2006 0m M m M 0 ( )h x , ( )h x ,
[ ]0 0 0 01, 2006 : ( ) 0 ( ) ( )x h x f x g x = = 3. f [ ],a
( ) ( ) 0f a f > . ( )0 ,x a ( )0 0f x = . [ ],x a ( ) ( ) 0f x f a .
[ ],x a ( ) ( ) 0f x f a , [ ], : ( ) ( ) 0a f f a <
[ ] ( ). .
1 1
( ) ( ) 0, : ( ) 0
,f f a
a ff a
< =
, [ ] [ ). .
2 2
( ) ( ) 0( ) ( ) 0, ( ) 0
,( ) ( ) 0f ff a f
fff f a
= <
f ( ),a , .
-
3
: . . 3
4. [ ]: ,f a \ : 2( )f > . : 3 33 ( )f x dx
< , 2( )f x x= .
3 3
3 :
32
3x x dx
= : 2( ) ( )g x f x x= [ ]0 0, : ( ) 0x a g x =
2 2( ) ( ) 0 ( ) 0f f g > > >
3 33 3
2 2
3 ( ) ( )3
( ) ( ( ) ) 0
( ) 0
af x dx a f x dx
f x dx x dx f x x dx
g x dx
< < < < .
f f f f f ( [ ],a )
( ) 0( )
( )f x
f x f x
-
4
: . . 4
( ) 0 ( ) ( )(1) ( ) ( ) ( ) ( ) (2)( ) ( )
f x f a ff a f f f af a f
= =
: [ ]( )( ) ,( )
f xg x af x
= ,g :
( )( )
( )2
2 2
( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) (3)( ) ( )
f x f x f xf x f x f x f xg x g xf x f x
= =
(2)
( )( )( )
( ) ( )( )( )( )
f ag af a
g a gfgf
= ==
[ ]( ) ( ). . 0 0
,
, , : ( ) 0( ) ( )
Rolleg
g x g xg a g
=
=
( ) ( ) ( )( ) ( )
20
2(3)0 0 0
0 20
( 02
0 0 0 0 0
0
( ( ) ( ), : 0
(
( ( ) ( ) ( ) ( ) 0f x
f x f x f x x a
f x
f x f x f x f x f xx
>
=
= >
6. f \ ( ) ( ) 0 (1)f x f x+ + = , x\ , * \ . ( ) 0f x = .
(1)
(1) ( ) ( ) 0 ( ) ( )
(1) ( ) ( ) 0 ( 2 ) ( )
( 2 ) ( ) (2)
x x
x x
f x f x f x f x
f x f x f x f x
f x f x
+ + + = = + + + + = + = +
+ =
( ) ( )2(1) 2 2 0 ( ) ( 2 ) 0( ) ( 2 ) 0 ( 2 ) ( ) (3)
(2), (3) ( 2 ) ( ) ( 2 ) (4)
x xf x f x f x f x
f x f x f x f xf x f x f x
+ + = + = + = =
= = +
( )f x 2T = . 0 (0) (2 )x f f = = ,
-
5
: . . 5
[ ]( ) ( ). .
0, 2
0, 2 0, 2 : ( ) 0(0) (2 )
Rollef
f ff f
=
=
(4) ( 2 )( 2 ) ( ) ( 2 )( 2 )( 2 ) ( ) ( 2 )
f x x f x f x xf x f x f x
= = + + = = +
f 2T = . , f ( )0, 2 , , 2 ( ).
( ) 0f x = , .
7. f [ ]1, 4 . (1) 1, (2) 2, (3) 3f f f= > < (4) 4f = , ( )0 1, 4x
0( ) 0f x = .
(1) 1(4) 4
ff
= = : ( ) ( )g x f x x= , ( ) ( ) 1g x f x =
(1) (1) 1 0(4) (4) 4 0(2) (2) 2 0(3) (3) 3 0
g fg fg fg f
= == == >= , \ : ( ) ( )f f = .
( )1 i A + , 1 1
11
1 1 1
11
: ( ) (1 ) ( )
( ) (1)( )
x x
xx
x e if x i e if x i
ee f x
f x
+ = + + = += ==
\
( )1 i A + ,
-
7
: . . 7
22 2
2 2 22
: ( ) (1 ) ( ) (2)( )
xx xex e if x i e f x
f x
= + = + ==\
1
1 2
2
1
1 2
2
( ) 1( ) ( )(1), (2)
( ) 1
x
x x
x
f xf x f xe
f x e ee
= =
=
: ( )( ) xf xg xe
=
2
( ) ( ) ( ) ( )( ) ( ) (3)( )
x x
x xf x e f x e f x f xg x g x
e e = =
1
2
11
1 22
2
( )( )( ) ( )
( )( )
x
x
f xg xe g x g x
f xg xe
= =
=
[ ]( )
1 2(3).
1 2 1 2
1 2
,( , ) , : ( ) 0 ( ) ( )
( ) ( )
Rolleg x xg x x g f fg x g x
= =
=\
10. :f \ \ ( ) ( ) 1 (1)f x f x = x\ . (0) 1f = , :
) ( ) ( ) 1f x f x = x\ ) ( ) xf x e= x\ ) fC : 1y x = +
) (1) ( ) ( ) 1 (2)x x
f x f x =
( )( )
(1), (2) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 0
( ) ( ) ( ) ( ) 0
( ) ( ) 0( ) ( )
f x f x f x f xf x f x f x f x
f x f x f x f x
f x f x xf x f x c
= =
+ = =
=\
0 (0) ( 0) 1x f f c c= = = , ( ) ( ) 1 (3)f x f x = , ( ) 0, ( ) 0f x f x
-
8
: . . 8
) 1(3) ( )( )
f xf x
=
1
01 1
1(2) ( ) 1 ( ) ( ) ( )( )
0 : (0) 1
xf x f x f x x f x C ef x
x f C e C
= = == = =
\
( ) xf x e=
) ( )0 0, ( )A x f x . : 00( ) xf x e= , 0
0( )xf x e = .
( )1 0 0 0 0 0 0 0: ( ) ( ) ( ) ( ) ( )y f x f x x x y f x x f x x f x = = + , 2 : 1y x = + . 1 2
0
0 0
0 00 0
0 0 0 0
( ) 1 1 0( ) ( ) 1 1 0 1 0 1 1
x
x x
f x e xf x x f x e x e e e = = = + = + = + = + =
1y x= + ( ) ( )00, 0,1e = 11. f [ ],a ( ), ( ) ( )f a f = . ( )1 2, ,..., ,a ,
1 2( ) ( ) ... ( ) 0 (1)f f f + + + =
i (1) [ ],a
.
: x + =
-
9
: . . 9
1 1
1 1 1
x
x
x
x x x
= = + =
= = =
f .
f ...
( )
1 1
2 2
3 3
( ) ( ), : ( )
( 2 ) ( ), 2 : ( )
( 3 ) ( 2 )2 , 3 : ( )
...
( ) (1 , : ( )
f a f aa a f
f a f aa a f
f a f aa a f
f f aa f
+ + = + + + + = + + + + =
+ + = ( )1 )
+
( )1 2
1 2
( ) ( ) ... ( )
( ) 2 ... ( ) 1
( ) ( ) 0( ) ( ) ... ( ) 0
f f f
f a f f f a f f a
f f af f f
+ + + = + + + + + + + =
+ + + = = =
-
10
: . . 10
12. f [ ],a , ( ) 0f = . ( ) 0f > , [ ],a f
.
f f f f f ( ) 0( ) ( ) ( )( ) 0 lim 0 lim 0
f
x x
f x f f xfx x
=
> > >
( ) 0f xx > .
( )0 : , x > ( ) 0f xx > .
( ),x ( ) 0f x < 0x < , . ( )( ) 0 ,f x x < f , [ ]( ) ,f x x 0 .
13. 1x
x xI dxe
= + . : i)
1
x
x
xe xI dxe
= + ii) I =
i) : ( )x y x a y = = + x y dx dy dy dx= = = x y yx y y
= = == = =
( ) ( )( ) ( ) 11 1 1x y yy yx x y yI dx dy dy
e ee
= = = =+ + +
1 1 1
y x
y y x
y
y y ye y xe xdy dy dxe e e
e
= = =+ + +
ii) 1 21
1
xx
xx
x
x xI dxx x xe xe I dx
exe xI dxe
+
= ++ = += +
-
11
: . . 11
( ) ( ) ( )( ) ( )( ) ( ) [ ]
(1 )2 21
2 2
2 ( ) ( )
2 ( 1) ( 1)
2 ( ) 2 2
x
x
x x eI dx I x xdxe
I x x dx I x x x dx
I xdx
I x
I I I
+ = =+ = =
= + = + + = + + = =
14. : /2
0
( )( ) ( ) 4
x
x xxI dx
x x
= =+ .
: 2
x y dx dy dy dx= = =
0 02 2
02 2 2
x y y
x y y
= = = = = =
2
2
x y y
x y x
= = = =
, ( )( ) ( ) ( )0
/2
y
y y
yI dy
y y
= +( )
( ) ( )/2
0
y
y y
yI dy
y y
= + ( )
( ) ( )( )
( ) ( )
( )( ) ( )
/2
0 ( ) /2
0/2
0
( )2
x
x x x x
x xx
x x
xI
x x x xI dx
x xx I dx
x x
+= + + = += +
/2
02 2 0
2 4I dx I I
= = =
-
12
: . . 12
15. f \ , 2( )f x x > x\ .
21( ) 12
f x x x= + , \ .
: 21( ) ( ) 12
g x f x x x= +
( ) 0g x = \ . 2 2( ) ( ) 0f x x x f x x x > > \ \
3 3
( ) 0 ( ) 03 3x xf x x f x x
> > \ \
3
( )3xf x \ .
( ) ( ) 1g x f x x = + 2 2 2( ) ( ) 1 1 ( ) 1f x x f x x x x g x x x > + > + > + 2 1 0x x x + > \ ( 0 < )
( ) 0g x x > \ , ( )g x / \ .
3
( ) ( ) 3 33 00 ( ) (0) ( ) (0)3 3
xh x f x xx h x h f x f=
> > > /
( )3( ) (0) 0,3xf x f x > + +
3
lim3xx
+ = + , lim ( )x f x+ = +
3 3( ) 00 ( ) (0) ( ) (0)
3 3
h x xx h x h f x f< < < / 3
( ) (0)3xf x f < +
3
lim3xx
= , lim ( )x f x =
( )g x / \ ( )( ) ( ) lim ( ), lim ( )x xg g f x f x + = =\ ( )g x \ ( )g = \
0 ( )g ( ) 0g x = \ ( )g x .
-
13
: . . 13
16. f [ ],a ( )Zx x i f x= + , ( )Wx x i f x= + , [ ],x a . Im( ) Im( )Za Z= , ( )1 2, ,a , 1 2 1 2W W W W + = + .
Im( ) Im( ) ( ) ( ) (1)Za Z f a f = = :
( ) ( )1 2 1 2
1 1 2 2 1 1 2 2
1 2 1 2
1 2 1 2
1 2 1 2
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
( ) ( ) ( ) ( )2 ( ) 2 ( ) 0 ( ) ( ) 0
W W W Wif if if if
f f i f f if f f f
f f f f
+ = + + + + = +
+ = + = + = + =
( )
( )
1 1. .
2 2
( ) ( )2, , : ( ) 2, , ,
22 2
( ) ( ), , ,22 2 , , : ( ) 2
2
af f aaa a a a ff a
aa a f ff a a a f
+ ++ + = ++ + + =
1
1 2
1
( ) ( ) ( ) ( ) ( ) ( )2 2( ) ( ) 2 2
( ) ( ) 02 2 0
a af f a f f f f af f
f f
++ + + + = = =
= = =
7. 0a > 1xx e a= , .
0 0x > , 01
0xx e a=
( )
( )
0 0
0 0
1/ 1/0 0
0 00 0
(ln 1)0 0
ln ln ln ln ln
1 lnln 1 ln ln 1
ln ln 1
x x
x x
x e a x e a
ax a xx x
a x x a e
= = + = + = = =
, 0 0(ln 1)x xe .
: (ln 1)( ) x xf x e = ( )0,fD = +
-
14
: . . 14
[ ](ln 1)
(ln 1) (ln 1)
(ln 1)
0(ln 1)
( ) (ln 1) (ln 1 1)
( ) ln
( ) 0 ln 0 ln 0 1x x
x x x x
x x
ex x
f x e x x e x
f x e x
f x e x x x
= = + =
( )
( )
1 ( ) (1)( ) (1)
1 ( ) (1)
f x
f x
x f x ff x f
x f x f
< >
/
0
1(ln1 1) 1 1min ( ) (1) min ( )f x f f x e ee
= = = = . 1min ae
= .
18. f ( )2 2 2 2( ) 36 9 ,f x a x a x x x= + + \ 0a . f 0x x= , :
) : ( ) 22 00 0 2722 18 9a xa x x a + = + ) 0 0x = , .
)
0 .
0 0
0
( ) 0 (1)Fermat
f xx f x
x
=\
( ) ( ) ( )( ) ( ) ( )
( ) ( )( )
0
2 2 2 2
2 2 2 20 0 0 0 0 0
12 2 2
0 0 0
22 0
0 0 2
( ) 72 2 9 9 9
( ) 72 2 9 9 9
0 72 2 9 9
722 189
x x
f x a x a x x a x x a
f x a x a x x a x x a
a x a x x a
a xa x xa
=
= + + + + = + + +
= + + + = +
) ( ) 220 272 00 2 0 18 0 0 09ax a a = + = =+ , .
-
15
: . . 15
( ) 22 00 0 22 2 2
0 202 2 2
722 18 1 19
72 9 91 0 9 09 72 72
a xa x xa
a x a ax a aa a a
+ ++ + + + \
19. a, b, c, d :
(1) 0a b c d + + < (2) 0d > (3) 0a b c d+ + + < 2 3b ac . (: 3 2( )f x ax bx cx d= + + + ( )1,0 ( )0,1 ).
3 2
2
( )( ) 3 2
f x ax bx cx df x ax bx c
= + + + = + +
( 1) 0( 1) (0) 0
(0) 0
(0) 0(0) (1) 0
(1) 0
f a b c df f
f df d
f ff a b c d
= + + < = >
[ ] ( )[ ] ( )
. .
1 1
. .
2 2
1,01,0 : ( ) 0
( 1) (0) 0
0,10,1 : ( ) 0
(0) (1) 0
ff
f f
ff
f f
= < = . ) ( ) 0
x
xf t dt = .
) 0
( ) ( )x
f t f t dt ( ) ( ) 0 ( )F x f x x F x = > \ / \
) 0 0
( ) 0 ( ) ( ) 0x x x
xf t dt f t dt f t dt
= + =
( ) ( ) 0 ( ) ( )0
FF x F x F x F x
x x x x x x
+ = = = =
/
\
0x =
-
17
: . . 17
( ) 0
( ) 0f tx
xf t dt x x
>= =
22. f : ( ) 32( ) 2 3 33x xf x x x e x= + + ) f :
( ) 0f x > x\ . ) ( , )M a =
2
2
2
4( ) 0
y y
x xf t dt
=
) 2 2( ) (2 2) ( 2 3) 1x xf x x e x x e x = + + 2 2 2
2
( ) ( 1) ( 1) ( ) ( 1) ( 1)( ) 0 ( 1) ( 1) 0 1 0
x x
x x
f x e x x f x e xf x e x e x
= + + = + +
( ) ( ],0 ( ) 0 ( ) ,0x f x f x < 0 ( ) [ )0, ( ) 0 ( ) 0,x f x f x + > +/
0 ( ) (0) ( ) (0) ( ) 6( ) 0
0 ( ) (0)
f
f
x f x f f x f x f x x f x x
x f x f
> < >
/
2\ \
\
) 2
2
( ) 02 2 2
4( ) 0 2 4
f ty y
x xf t dt y y x x
> = =
( )
2 2
2 2
4 2 04, 2, 0
4 16 4 20 0
, 2,12 2
20 52
x y x yA BA B
A BK K
R R
+ == = =+ = + = >
= =
-
18
: . . 18
23. 1) :f \ \ : ( ) ( ) ( ) xf x f x f x ce = = , c .
2) ) \ f :
( )2 21 ( )( ) 1 1 (1)1x f tf x x dt xt = + + + \ ) f
(c) .
) (c)
.
1) ( ) ( ) ( ) ( ) 0 ( ) ( ) 0x x xf x f x f x f x f x e f x e e = = =
2 2
( ) ( ) ( ) 0 ( ) 0( ) ( )
( ) ( )
x x
x x x
xx
f x e f x e f x xe e e
f x c f x c e xe
= = = =
\
\
: ( ) ( ) ( )xf x ce f x f x = =
2) 2
2 21
21
( )( ) ( ) 1(1) 1
( )1 11
x
x
f t f x f t tdtf tx t dt
t
+ = + + + +
2 2 2 21
(1)
2
1 1 1
1 1 22
( ) ( ) ( ) ( )11 1 1 1( )
1(1) 1 (1) (1 1)(1 0) (1) 2
(1) 21 1 2
( ) ( ) ( 1)1
x
x
x x
f x f t f x f xdtx t x x
f x c exx f ff c e c e c e
f x e e f x e xx
= + = + + + + = +
= = + + == = =+= = ++
1 2 1 1 2
1 2
( ) ( 1) 2 ( ) ( 1 2 )( ) ( 1) 0
x x x
x
f x e x e x f x e x xf x e x x
= + + = +
= \
-
19
: . . 19
1x = , ( )f x / \ .
1 2 1 1 2
1 2
2 2
( ) ( 1) 2( 1) ( 2 1 2 2)( ) ( 1)
( ) 0 1 0 1 1 1
x x x
x
f x e x e x e x x xf x e x
f x x x x x
= + = + +
=
( )( )
1, ( 1) .
1, (1) .
f
f
: ( ) ( 1) ...B AA AB A
y yAB y y x x y fx x = =
.. , f , ( ) 0fC AB f x AB
( )1 11 1
( ) ( ) ...E f x AB dx f x AB dx = = =
24. f : 1
( )tx ef x dt
t= ,
0x . 1) x ( ) lnf x x .
2) ( ) : 1
limt xx
x
e dtt
+ .
1) : ( ) ( ) lng x f x x= . ( ) 0g x . 1 1 1( ) ( )
x xe eg x f xx x x x
= = = (1) 0g = ( ) 0 0g x x
1 ( ) (1) ( ) 01 ( ) (1) ( ) 0
x g x g g xx g x g g x < < <
( )g x 1x .
-
20
: . . 20
2) 11 1 1
( )lim lim lim lim lim
txt x t tx x xx x
x xx x x x x
e dte e e f xtdt e dt e dtt t t e e
+ + + + += = = =
1 1
1 ( ) ln
t t tx xe e e dt dt t t t
x f x x
lim ln lim ( )
x xx f x + += + = +
, ( ) ( ) 1lim lim lim lim 0( )
x
x x xx x x x
ef x f x xe e e x
+ + + += = = =
25. f [ ]0,1 , ( )1 1 10 0 0( ) ( ) ( )f x f y dy dx f x dx .
( ) ( )1 1 1 1 1 10 0 0 0 0 01 1 1
0 0 01 1 1 1
0 0 0 02
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ( ) )
f x f y dy dx f x dx f x f y dy dx f x dx
f y dy f x dx f x dx
f x dx f x dx f x dx f x dx
= == == = == =
1 0a = > , : 2( 1) 4 1 0 1
4 4 1 4a = =
26. f \ 3
( ) ( )x
xg x f t dt
+= , g \ .
3
0 0( ) ( ) ( )
( ) ( ) ( 3)
3 ( ) ( 3) ( 3) ( ) 0( ) 0 , ( )
x x
f
g x f t dt f t dt
g x f x f x
x x f x f x f x f xg x x g x
+= + = + +
< + > + + < <
2
\ 0 \
-
21
: . . 21
27. ( fC ) f
x 1 . ( fC ),
xx 1x = x a= : 2 1 2E a= + , 1a > .
1) f .
2) 1
lim ( )x
xxf t dt
++ .
1) ( )2 2 2 21 111 2 1 1 1a a a xE a E x x dx dxx = + = + = + = + :
2( )
1xf x
x= + \ .
2
2
1lim ( ) lim lim 11 1 01 1
x x x
x xf xx x
x+ + += = = =++ +
1y = +
2 2
1lim ( ) lim lim 11 11 1
x x x
xf xx
x x = = = + +
1y = .
2)
-
22
: . . 22
( )( )( ) ( )
( ) ( )
11 1 2 2
2 2 2 2
2 2 2 2
2 2
2 2
2 2
lim ( ) lim 1 lim 1
lim 1 1 1 lim 2 2 1
2 2 1 2 2 1lim
2 2 112
2 2 1lim lim2 2 11 1
xx x
x xx x x x
x x
x
x x
f t dt t dt t
x x x x x
x x x x x x
x x x
xx x x x
xx x x
++ ++ + +
+ +
+
+ +
= + = + = = + + + = + + + =
+ + + + + + += =+ + + +
++ + = = + + + +
2 22 2 11 1
2 0 2 121 0 0 1 0
xx x x
= + + + + += = =+ + + +
( )f t [ ], 1x x + ( )
( ) ( )[ ]
22 2 22 22
2 2 2 2 2
1 1 11( ) 01 1 1 1 1
( ) , 1
tt t t t t ttf tt t t t t
f t x x
+ + + + = = = >+ + + + + +/
- ( ) (1)m f t M , ( )m f x= , ( 1)M f x= +
( ) ( )1 1 1
1
1 1
(1) ( )
1 ( ) 1
( ) ( ) ( ) ( 1) (2)
x x x
x x xx
xx x
x x
mdt f t dt Mdt
m x x f t dt M x x
m f t dt M f x f t dt f x
+ + +
+
+ +
+ + +
( )
2 (2) 1
. .
2
lim ( ) lim ... 11
lim ( ) 11lim ( 1) lim 11 1
x xx
xx
x x
xf xx
f t dtxf xx
+ + ++
+ +
= = =+ =++ = =+ +
28. f \ ( )f x x < , x\ . : (4) (2) 6f f < .
-
23
: . . 23
2 2
( ) 0 ( ) 0 ( ) 02 2x xf x x f x f x
< <
-
24
: . . 24
(0) 3f = , 3c = : 21( ) 3 32
f x x x= + , 2x .
, f 2 , : 2
(2) lim ( )x
f f x= . , : ( ) ( )1, 2 2,3x , 2x
2
2 2
1(2) lim ( ) lim( 3 3) 12x x
f f x x x = = + = . : 21( ) 3 3
2f x x x + ,
x\ . , .
30B. f \ : ( ) 0f x
( )1 0 ( ) 2 2x u f t dt du x , x\ . fC , xx 0x = 1x = .
: ( )1 0( ) ( ) 2 2x ug x f t dt du x= + (1) 0g =
0( ) ( ) 2 (1)
xg x f t dt = ( ) 01 1 1
0 0 0( ) ( ) ( ) (2)
f xE f x dx f x dx E f t dt
= = = (1). 1
0
1 .1 ( ) (1) (1) 0 ( ) 2 0
1
Fermatg g g x g g f t dt
= =
\
(2)
2 0 2 . .E E = =
31. 4
1( ) 41
f xx
= ++ 0x > .
) f .
) 1
lim ( )x
xxf t dt
++ .
-
25
: . . 25
) 3
4
4( ) 0 02 1
xf x x fx
= > > + /
) ( )f t [ ], 1x x + f / - ( ( )m f x= ( 1)M f x= + )
( ) ( )1 1 1
1
1
( ) ( )
1 ( ) 1
( ) ( ) ( 1)
x x x
x x xx
xx
x
m f t M mdt f t dt Mdt
m x x f t dt M x x
f x f t dt f x
+ + +
+
+
+ + +
. . 1lim ( ) ...lim ( )
lim ( 1) ...xx
xxx
f xf t dt
f x
+++
+
= = + = ++ = = +
27
( )f x .
32. f 0 1x = , :
( ) ln 1 (1)f x x x , 0x > . : 1(1)2
f = .
1
f , : ( ) ( ) ln 1g x f x x x= +
(1) 0g = 1 1( ) ( ) ln ( ) (2)2
g x f x x f xx x
= +
( ) (1) 0g x g x >
( )(2).
11 1 11 (1) 0 (1) ln1 (1) 0 (1)1 2 2
1 0,
Fermatg
g f f f
= + = =
+
2
f , Fermat.
1 ln ln1 ln 0x x x> > >
-
26
: . . 26
00
1 1
1 1 1
( ) ln 1 1(1) ( ) (3)ln ln ln
11 12lim lim 1ln 2
1 1(3) lim ( ) lim lim ( ) (4)ln 2
x x
x x x
f x x x xf xx x x
x xx
xxf x f x
x+ + +
= =
1 ln ln1 ln 0x x x< < <
1 1 1
( ) ln 1 1(1) ( )ln ln ln
1 1lim ( ) lim lim ( ) (5)ln 2x x x
f x x x xf xx x x
xf x f xx
f 11 1
lim ( ) lim ( ) lim ( ) (1)xx x
f x f x f x f + = = = 1(4) (1) 12 (1)1 2(5) (1)2
ff
f
=
33. 2 2 21 2 2 3 1 3 (1)z z z z z z + = , 3 .
1 2 3( ) ( ) ( )A z B z z 2 2 2(1) + =
, , , , .
34. f , : , (1) 1 , ,f f z x = \ \ ^ \
( )2 11 1
2 5 ( ) 5 12 1 (1)x x tz i f t dt z i e dt x+ + +
) ( ) ( )M z c .
) ( )h x ( )c . )
1( ) ( )
xH x h t dt= , , , 1xx yy x = .
-
27
: . . 27
) ( )2 11 1
( ) 2 5 ( ) 5 12 1x x tg x z i f t dt z i e dt x= + + +
( )22
1 2
1
(1) 0
( ) 2 5 ( ) 5 12
( ) 2 5 ( ) 2 5 12 (2)
x
x
g
g x z i f x z i e x
g x z i f x z i x e
= = + + +
= + + +
(1) ( ) (1)g x g gC 1
. Fermat 2(2) 1 1(1) 0 2 5 (1) 2 5 1 12 0g z i f z i e = + + =
( ) ( )( ) ( ) ( )
5 5 6 5 5 6
0 5 0 5 6
0, 5 0,5
z i z i z i z i
z i z i
E E M z
+ + + = + + = + + =
6ME ME + = 2 6 3a a= =
( ) ( ) ( )
2 2 2 2 2 2
2 2 2 2
91 19 16 9 16 16 9
16 16 169 9 99 9 9
x y x y y x
y x y x y x
+ = = =
= = =
( )216( ) 99h x x=
) 1
0( )E H x dx=
[ ]
( )
1
1 1 11
00 0 01 1
1 00
1 0 1 1 1/22 2
1 1 0 0
*1/2
( ) ( ) ( ) ( ) 0 ( ) 0 ( )
1 ( ) (1) ( ) 0
( ) ( ) ( ) ( )
( ) ( )
4 41 ( ) 0 ( ) 9 93 3
2 23
x
x
H x h x dt H x h x H x H x
x H x H H x
E H x dx x H x dx xH x xH x dx
x h x dx xh x dx
h x dx h x dx x x dx x x dx
u
= = > > > > >
= = = = = =
= = =
=
/
8 8 81/2
9 9 9
2 2 ......3 3
xdx u du = =
-
28
: . . 28
2 9 2* 0 9
1 8
u x du xdxx ux u
= = = = = =
35. ( ): 1, , :f f + \ ( ) ( )0
2
4 2 ln 1 (1), 1x tf x t dt x x = + > . ) : ( )2
1( )1
f xx
= +
) g me ( )3 2( ) ( )g x x x f x= + , , , 1x x y y x = ) ( ) ( )x xh x g e e =
) 12 22
u x t du dt dt du= = =
( )
( )
0
02
0 0 0
0 0
(1)
0 0
0
0
2 02 20 2 0
4 12 ( )2 2
( ) ( ) ( )4 4 4
( ) ( )4 4
( ) ( ) ln 1
1( ) ( ) ( )1
( )
xx
x x x
x x
x x
x
x
x xt u x u
t u x u xxtf x t dt f u du
u x u xf u du f u du f u du
u xf u du f u du
uf u du x f u du x
xf x f u du xf xx
f u du
= = == = =
= = = = =
= + = +
2
1 1 1( ) ( )1 1 ( 1)
f x f xx x x
= = = + + +
) ( ) ( )23 2 2 21( ) ( ) ( 1) 0 1,( 1) 1xg x x x f x x x x x = + = + = > ++ +
-
29
: . . 29
( ) ( ) ( )
2 21 1 1
0 0 0
121
00
1 1( )1 1 1
1 ln 1 ln 1 ...2
x xE g x dx dx dxx x x
xx x dx x x
= = = + = + + + = + + = + + =
) + ( )2 0
0
0
lim ( ) lim ( ) lim1
lim lim1 1
1lim lim 111 11
, lim 0
lim lim 1
xx x x
xx x x
x x xx x
x x xx x
x x
xx x xx
x x
xx
xx u
eh x g e e ee
e e ee e le e ee e
e ee
u e u e
e ue u
+ + + +
+ +
+ +
+
+
= = =+= = =+ +
= = =+ + = = =
= =
1l = , lim ( ) 1x
h x+ = 1y = + .
36. :f \ \ 3 2( )f x x x x= + , x\ . i) f 1-1 ( )1,1A f 1f .
ii) fC ( )1,1A fC .
iii) 1f
( )1,1A . iv) :
1
1
( ) 1lim1x
f xx
.
i) 2( ) 3 2 1 0f x x x = + > x\ , 23 2 1x x + 4 12 8 0 = = < . , f 1-1. ,
-
30
: . . 30
(1) 1f = , 1(1) 1f = . ( )1,1A f 1f .
ii) ( ) fC ( )1,1A : (1) (1)( 1) 1 2( 1) 2 1y f f x y x y x = = =
( ) 2 1f x x= :
( ) ( ) ( )( )3 2 3 2
2 2
2 1 1 0
1 1 0 1 1 0
1 1
x x x x x x x
x x x x x
x x
+ = + = = = = =
( ) fC : ( )1, 3B iii) 1,f fC C y x= . ( ) fC ( )1,1A ( ) 1fC ( )1,1A .
( ) : 2 1y x= ( ) : 1 12 1
2 2x y y x= = +
iv) 1 1
1
( ) (1)lim1x
f x fx
1f 0 1x = ,
( ) , 12
= .
, 1 1
1
( ) (1) 1lim1 2x
f x fx
=
37. [ ): 0,f + \ . i) : ( ) ( 1) ( ) ( 1)f x f x f x f x < + < + , 0x > . ii) [ ), : 0,F G + \ :
1( ) ( ) ( )
x
xF x f t dt f x
+= 1( ) ( ) ( 1)xxG x f t dt f x+= + , 0x . iii) :
2 1
1 0(1) (0) ( ) ( ) (2) (1)f f f t dt f t dt f f < <
-
31
: . . 31
i) f [ )0,+ , ... [ ], 1x x + 0x > .
( ) ( 1) ( ), 1 : ( ) ( 1) ( )( 1)
f x f xx x f f x f xx x
+ + = = + +
( ) ( ) ( 1)f x f f x < < + , 0x > 1x x< < + . f / f .
ii) 1 1 1
1 1 1( ) ( ) ( ) ( ) ( ) ( ) ( )
x x x
xF x f t dt f t dt f x f t dt f t dt f x
+ += + = 1
1 1( ) ( ) ( ) ( 1)
x xG x f t dt f t dt f x
+= + , ( ) ( 1) ( ) ( )F x f x f x f x = + ( ) ( 1) ( ) ( 1)G x f x f x f x = + + 0x .
( ) 00
( ) 0F x
xG x
> > <
,F G [ )0,+ , F : G :
iii) ,F G : (0) (1)F F< (0) (1)G G> 1 2
0 12 1
1 0
(0) (1) ( ) (0) ( ) (1)
(1) (0) ( ) ( )
F F f t dt f f t dt f
f f f t dt f t dt
< < <
2 1
1 02 1
1 0
(1) (0) ( ) (2) ( ) (1)
( ) ( ) (2) (1)
G G f t dt f f t dt f
f t dt f t dt f f
< < <
: 2 1
1 0(1) (0) ( ) ( ) (2) (1)f f f t dt f t dt f f < < .
-
32
: . . 32
38. :f \ \ , : 20
1( )( ) 1
xf x dt
f t= + , x\ .
i) f .
ii) 0
( )limx
f xx
.
iii) : ( )3( ) 3 ( ) 3f x f x x+ = x\ . iv) f ( )f =\ \ , 1f 0, 1y x= = .
i) 21
( ) 1f t + \ 201
( ) 1x
f t + \ .
2 20
1 1( ) 0( ) 1 ( ) 1
xf x dt
f t f x
= = > + + x\ . , f .
ii) f 0
200
1lim ( ) (0) 0( ) 1x
f x f dtf t
= = =+ ( 00 :) de L 'Hospital
2 20 0 0
( ) ( ) 1 1lim lim lim 1( ) ( ) 1 (0) 1x x x
f x f xx x f x f
= = = = + +
iii) 21( )
( ) 1f x x
f x = + \
2 ( ) ( ) ( ) 1f x f x f x + = ( ) ( )2 33 ( ) ( ) 3 ( ) 3 ( ) 3 ( ) 3f x f x f x f x f x x + = + = 3 ( ) 3 ( ) 3f x f x x c+ = + x\
30 : (0) 3 (0) 0x f f c c= + = = , 3 ( ) 3 ( ) 3f x f x x+ = x\
iv) f / 1-1. 1f \ . 3 31( ) 3 3
3f x y y y x x y y= + = = +
, 1 3 21 1( ) 13 3
f x x x x x = + = + x\ .
-
33
: . . 33
1( ) 0f x [ ]0,1x . 13 4 21 11
0 00
1 1 7 7( ) . .3 12 2 12 2 12 12x x xE f x dx x dx E = = + = + = + = =
39. :f \ \ , : 20( )( ) 2
1x f uf x x duu
> + x\ . :
i) 202
( )1( )1
x f u duug xx+= +
, x\ .
ii) 20( ) 0
1x f uu
>+ 0x > . iii) ( ) 0f x > 0x > .
iv) 2 1
2 20 0
( ) 1 ( )1 2 1
x f u x f udu duu u
+>+ + , 1x > . v) lim ( )
xf x+ , + .
i) ( )
( )2
2 20 0
22
( ) ( )1 21 1( )
1
x xf u f ux du x duu ug x
x
+ + + = =+
( ) ( )( ) ( )
222 0 20
2 22 2
( ) ( ) ( )1 2 ( ) 211 1 01 1
xxf x f u f ux x du f x x duux u
x x
+ ++ += = >+ +
, x\ .
, g .
ii, iii) 0x > : ( ) (0)g x g>
202
( )1 01
x f u duux+ >+
, 2 1 0x + > 20
( ) 01
x f u duu
>+ 0x > : 2 0x > 20
( ) 01
x f u duu
>+ , 20
( )( ) 2 01
x f uf x x duu
> >+ , 0x >
-
34
: . . 34
iv) g/ 1x > : ( ) (1)g x g>
2 102 20
( )1 ( )1
1 2 1
x f u du f uu dux u+ >+ +
1x > :
2 1
2 20 0
( ) 1 ( )1 2 1
x f u x f udu duu u
+>+ +
v) (iv) : ( ) 122 20 0( ) ( )2 11 1x f u f ux dt x x duu u> ++ + , 1x > . ( ) 13 20 ( )( ) 1f uf x x x dtu> + + , 1x > (1)
, ( )3 3lim limx x
x x x+ ++ = = + 1
20
( ) 01
f u duu
>+ , ( ) 13 20 ( )lim 1x f ux x duu+ + = ++
( ) 13 20 ( )lim ( ) lim 1x x f uf x x x duu+ + + + , lim ( )
xf x+ = +
40. f , \ , [ ],a , 0a < < . : i) 1 2,x x , 1 2x x , 1 2( ) ( ) 0f x f x = = . ii) 3x \ , 3( ) 0f x = . iii) ( ) ( ) ( ) 0f x f x f x + = \ . iv) [ ]2( ) ( ) 0f x f x + = , \ .
i) f [ ],a , , 1x \ 1( ) (1)f x a= , 2x \ 2( ) (2)f x = . f \ , Fermat ( )1 0f x = ( )2 0f x = . , 1 2x x .
-
35
: . . 35
ii) .. 1 2x x< . f [ ]1 2,x x ( )1 2,x x 1 2( ) ( ) 0f x f x = = . Rolle, ( )3 1 2,x x x 3( ) 0f x = .
iii) ( ) ( ) ( ) ( )g x f x f x f x = + [ ]1 2,x x : 1 1 1 1( ) ( ) ( ) ( ) 0g x f x f x f x a = + = < 2 2 2 2( ) ( ) ( ) ( ) 0g x f x f x f x = + = >
Bolzano ( )1 2,x x ( ) 0g = . .
iv) ( )( ) ( ) f xh x f x e= [ ]1 2,x x ( )1 2,x x ( )2( ) ( ) ( )( ) ( ) ( ) ( ) ( )f x f x f xh x f x e f x e e f x f x = + = +
1( )1 1( ) ( ) 0
f xh x f x e= = 2( )2 2( ) ( ) 0f xh x f x e= = Rolle, ( )1 2,x x , : ( )2( ) 0 ( ) ( ) 0h f f = + =
41. ( )f x ( )0, + ( ) 1f x 2
21 1( ) 12
x
x
tf x f dtx x = + , 0x >
1( )( ) 1
g x xf x
= + , 0x > .
:
i) f .
ii) g .
iii) f ( )1
xf xx
= + , 0x > ( )f . iv)
0
( )limln( 1)x
f xx + .
i) : t u t xux= = dt xdu=
1t x u= = 2t x u x= =
-
36
: . . 36
[ ] [ ]2 21 1
1 1 1( ) 1 ( ) 1 ( )2 2
x xf x f u xdu f u du
x= + = +
, f : [ ] [ ]2 21
1( ) 1 ( ) 1 ( ) 0 (1)2
xf x f u du f x
= + = > , f .
ii) g : [ ](1)
21 ( )( ) 1 1 1 0
( ) 1 ( ) 1f xg x x
f x f x
= + = + = + =
iii) 1x = [ ]211
1 1(1) 1 ( )2 2
f f u du= + = 1 1(1) 1 1 2 1 11(1) 1 1
2
gf
= + = + = + = g , :
( ) 1g x = , 0x > . 1 1( ) 1 1 1 ( )
( ) 1 ( ) 1 1xg x x x f x
f x f x x= + = = = +
, 0
lim 01x
xx+
=+ lim 11xx
x+=+
, ( ) ( )00( ) lim ( ), lim ( ) 0,1xxf f x f x+ = = iv) ( )
( )20 0 0 0 0
11( ) 111lim lim lim lim lim 11ln( 1) ln( 1) 1ln( 1)1
x x x x x
xxxf x xx
x x xxx
++ += = = = =+ + ++ +
42. f , \ , : 22( )f x x z x z= + , z z z i= + , , , 0a a \ . ) lim ( ), lim ( )
x xf x f x+ .
) f , 1 1z z+ > . ) f .
-
37
: . . 37
( ) ( )2 22 222 2 22 2 2 2 2 2
4( )x a x a axf x x z x z x a i x a i
x a x a
+ + + = + = + = = + + + +
) 24 4lim ( ) lim lim 0
x x x
ax af xx x+ + +
= = =
2
4 4lim ( ) lim lim 0x x x
ax af xx x
= = =
) ( ) ( )2 22 21 1 1 1 1 1 0z z a i a i a a a + > + + > + + + > + >
f \ : ( ) ( ) ( )( )2 2 2 2 2 2
22 2 2( ) 4
x x a x x af x a
x a
+ + + + = + +
( )( )
( )2 2 22 2 2
2 22 2 2 2 2 24 ( ) 4 , 0
x ax aa f x a ax a x a
+ + + = = >+ + + +
( ) 2 22 2 2 22 , 0a aM f a Ma += + = >+ f :
( ) ( ) ( ) ( ) ( )( ] [ ] [ ) [ ]
2 2 2 2 2 2 2 2lim ( ), , , lim ( )
0, , ,0 ,x x
f f x f a f a f a f a f x
M M M M M M
+ = + + + + = = = \
f .
) 0a > : ( ) [ ] ( ) 2 2 2 22 2 2 22 2, ,a a a af M M f a a
+ += = + +
\ \
0a < : ( ) 2 2 2 22 2 2 22 2,a a a af a a
+ += + +
\
2 2 2
4( ) 0 0 0axf x xx a = = =+ + , f 0x = .
-
38
: . . 38
43. :f \ \ ( )20
11
f xdt x
t=+ ,
x\ . ) :
i) f 0 0x = (0) 0f = . ii) f \ .
) \ ( )( )( )( )2 2
23
1 01
fof
fofdt
t
+ >+
) f \ , (0) 1f = .
) i) (0)
20
10 01
fx dt
t= =+
2
1 01 t
>+ t\ , (0) 0f = .
(0) 0f > , (0)20
1 01
fdt
t>+ ,
(0) 0f < , (0)20
1 01
fdt
t+ + (1) 2( 2) (3 )f f + > f
2 22 3 3 2 0 1 2 + > + > < >
-
39
: . . 39
) ( )0 0 0
20
( ) (0) ( ) ( )lim lim lim 101
f xx x x
f x f f x f xx x dt
t
= =+
( )f x u= , 0
lim ( ) (0) 0x
f x f = = , 0u , 02
lim (2)11
uu
o
u
dtt
+
20
1( ) ,1
ug u dt u
t= + \ ,
0
2 20 00 0
1 1lim ( ) (0) lim 01 1
u
u ug u g dt dt
t t = = =+ +
(2) 00
De LHospital :
0 0 0
2 2020
( ) 1lim lim lim 11 11
1 11
uu u uu
u u
dtdtt u
t
= = = + ++
, (0) 1f = .
44. f \ (0) 1f = 2 1( ) 3 ( ) 1f x f x = + ,
x\ . . :
i) x\ : 3 ( ) ( ) 2f x f x x+ = + . ii) f 1f :
iii) f 0 2x = ( 2) 0f = . . f .
.
f , ,x x y y 2x = .
. i) 2 321( ) 3 ( ) ( ) ( ) 1 ( ) ( ) ( )
3 ( ) 1f x f x f x f x f x f x x
f x = + = + = +
3 ( ) ( )f x f x x c + = +
-
40
: . . 40
( )30 : ( ) ( ) 2 2x f x f x x c= + = + =
ii) ( )y f x= : 3 32 2,y y x x y y y+ = + = + \ . x, f 1-1 ,
: 1 3( ) 2,f x x x x = + \ .
iii) 1(0) 2f = : ( 2) 0f = .
. 21( )
3 ( ) 1f x
f x = + \ ,
: 226 ( ) ( )( )3 ( ) 1
f x f xf xf x
= +
( ) 0f x > f ( )f x . ( )f x -2. :
2 : ( ) ( 2) ( ) 0 ( ) 0x f x f f x f x< < < > 2 : ( ) ( 2) ( ) 0 ( ) 0x f x f f x f x> > > <
f ( )2,0 .
. [ ]0 02 2
( ) ( ) 2 ( ) 0E f x dx f x dx x f x = = : 1( ) ( )f x u x f u= = , : ( ) ( )1 3 2( ) 2 3 1dx f u du u u du u du = = + = + (0) 1f = , ( 2) 0f = . :
( ) 14 21 12 30 0
0
3 5(3 1) 3 . .4 2 4u uE u u du u u du = + = + = + =
-
41
: . . 41
45. f , 5( ) 5 ,f x x x a a= + \ .( ) ) f .
) f .
) ( ) 0f x = , 4 4a < < .
) 4( ) 5 5f x x = 4 4 4( ) 0 5 5 0 5 5 1 1f x x x x x = = = = =
f ( ], 1 [ )1,+ , f [ ]1,1 .
f 1 1x = : 5( 1) ( 1) 5( 1) 1 5 4f a a a = + = + + = + f 2 1x = : 5(1) 1 5 1 4f a a= + = .
) ( ]1 , 1 = f ( ) ( )5 5lim ( ) lim 5 lim
x x xf x x x a x = + = =
( 1) 4f a = + , ( ]1( ) , 4f a = + . ( )2 1,1 = f ( )
1lim ( ) ( 1) 4x
f x f a f + = = + ( )
1lim ( ) (1) 4x
f x f a f = = , ( )2( ) 4, 4f a a = + . [ )3 1, = + f (1) 4f a=
( ) ( )5 5lim ( ) lim 5 limx x x
f x x x a x+ + += + = = +
-
42
: . . 42
, [ )3( ) 4,f a = + . ( )( ) ,f A = +
) 4 0 4a a < < + , 10 ( )f , 20 ( )f 30 ( )f ( ) 0f x = 1 2, 3 . , ( ) 0f x = 3 \ .
46. f , \ 5 3( ) ( ) ( )f x f x f x x+ + = , x\ .
) f . ) f 1f . ) fC ( ),1A a ( ), 2B , fC x a= x = .
) : 4 25 ( ) ( ) 3 ( ) ( ) ( ) 1f x f x f x f x f x + + = 4 2
4 2
1( ) 5 ( ) 3 ( ) 1 1 ( ) 05 ( ) 3 ( ) 1
f x f x f x f xf x f x
+ + = = > + + ( )4 25 ( ) 3 ( ) 1 0,f x f x x+ + > \ . , f .
) f \ , f 1-1, f . x 1( )f x , : 5 3( ) ( ) ( )f x f x f x x+ + =
( ) ( ) ( )5 1 3 1 1 1 5 3 1( ) ( ) ( ) ( ) ( )f f x f f x f f x f x x x x f x + + = + + = , 1 5 3( )f x x x x = + + .
) ( ) ( )f
f a f a < , [ ],x a .
( ) ( )2 24 2 5 31 1
26 4 2 6 4 2 6 4 2
1
( ) 5 3 1 5 3
5 3 5 2 3 2 2 5 1 3 1 16 4 2 6 4 2 6 4 2
320 5 3 1 78312 2 . .6 6 4 2 12
E f x dx u u u du u u u du
u u u
= = + + = + + = = + + = + + + + =
= + + =
-
43
: . . 43
47. ) *,a z \ ^ z a z ai = + , Re( ) Im( )z z= . ) f [ ], ( ) ( )z f i f = + , ( )0 ,x , ( )0 0f x = .
) f ( ), 0x [ ], , fC .
) : z x yi= + , ,x y\
( ) ( )2 22 22 2 2 2 2 22 2 2 2 Re( ) Im( )
z a z ai x yi a x yi ai x a y x y a
x ax a y x y ay a ax ay x y z z
= + + = + + + = + + + + = + + + = = =
) ( ) ( )f f = ( ) f [ ], 2( ) ( ) ( ) 0f f f =
( ) 0f = , ( ) 0f = , ( ) ( )f f = . , f , ( ) ( ) 0f f < . Bolzano: ( )0 0, : ( ) 0x f x = .
) ... [ ]0, x [ ]0 ,x ( )1 0,x x , : 01
0
( ) ( ) 0 ( ) ( ) 2 ( )( ) (1)
2 2
f x f f f ff xx
= = = =+
( )2 0 ,x x , : 02
0
( ) ( ) ( ) 0 ( ) 2 ( )( ) (2)
2 2
f f x f f ff xx
= = = =+
o f [ ] ( )1 2, ,x x o 1 2( ) ( )f x f x =
-
44
: . . 44
Rolle: ( ) ( )1 2, , : ( ) 0x x f = , fC .
48. f \ , f y y ( )0,3A : 2( ) ( ) ( ) x xf x f x f x e e = + , x\ . ) ( ) 2xf x e= + . ) f 1f .
) 1fC , x x x e= 2x e= + .
) ( )2 22 ( )( ) ( ) ( ) ( )2 2x
x x xf x ef x f x f x e e f x e = + = +
2 2 2 2.
2 2
( ) ( )( ) ( )2 2 2 2
( ) 2 ( ) 2 2
x xx x
x x
f x e f x ef x e f x e c
f x f x e e c
= + = + + = + +
2 0 0 20 : (0) 2 (0) 2 2 3 2 3 1 2 2 3 3 2 0x f f e e c c c c= = + + = + + = + = , ( )2 2 2 2( ) 2 ( ) 2 ( ) 2 ( ) 2 0x x x xf x f x e e f x f x e e = + + =
( ) ( ) ( ) ( ) ( ) 222 2 2 22 4 1 2 4 4 8 4 2 1 4 1 2 1 0x x x x x x x xe e e e e e e e = = + + = + + = + = + > ( ) ( )2 2 1( ) 1 12
xx
ef x e
+= = +
, ( ) 2xf x e= + ( ) xf x e= , (0) 3f = ( ) 2xf x e= +
) ( ) 2xf x e= + ( ) 0xf x e = > , f \ , f 1-1, f .
( )2 2 ln 2 , 2x xy e e y x y y= + = = > , 1( ) ln( 2), 2f x x x = >
-
45
: . . 45
) 1( ) 0 ln( 2) 0 2 1 3f x x x x > > > >
( ) ( )( ) ( )
( ) ( ) [ ] [ ]
2 3 21 1 1
3
3 2
3
3 2
323 23
333 2
3
( ) ( ) ( )
ln( 2) ln( 2)
2 ln( 2) 2 ln( 2)
2 ln( 2) 1 2 ln( 2) 1
0 2 ln 2 0
e e
e ee
e
e
ee e
e e
e
e
E f x dx f x dx f x dx
x dx x dx
x x dx x x dx
x x dx x x dx
e e x e x
+ +
+
+
+ +
+
= = + == + =
= + == + + = = + +
( ) ( )2 ln 2 4 . .e e e
== +
49. f [ ],a , [ ]2,3 ( ) 2f a = , ( ) 1f = . ) ( )0 ,x a , : 0( ) 0f x = . ) f ( ),a , i) fC . ii) f ( ),a , ( ),a , : 2012( ) ( ) ( ) 0f f f + = . iii) ( )1 2, ,a , 1 2 , :
1 2
1 1( ) 2 ( ) 2
af f
= .
) -
min max2 ( ) 1 ( ) 2 3f f f a f = < = < = < = , : ( )1 1 min, : ( ) 2x a f x f = = ( )2 2, : ( ) max 3x a f x f = =
Bolzano [ ]1 2,x x [ ] ( )2 1, ,x x a , ( )0 0, : ( ) 0x a f x =
) i) 1 2,x x ( ),a f 1 2,x x
f 1 2,x x
Fermat: 1 2( ) ( ) 0f x f x = =
-
46
: . . 46
, fC ( )1 1, ( )A x f x ( )2 2, ( )B x f x .
ii) g , 2012( ) ( ) ( ) ( )g x f x f x f x = + g [ ]1 2,x x [ ] ( )2 1, ,x x a ( )201220121 1 1 1( ) ( ) ( ) ( ) 2 0 2 0g x f x f x f x = + = +
Bolzano: ( ),a , : 2012( ) 0 ( ) ( ) ( ) 0g f f f = + =
iii) ... [ ]0,a x [ ]0 ,x ( ) 0 01 0 1
0 0 1
( ) ( ) 0 2 1, : ( ) (1)( ) 2
f x f a x aa x fx a x a f
= = =
( )2 0 , :x 0 0
20 0 2
( ) ( ) 1 0 1( ) (2)2 ( ) 2
f f x xfx x f
= = =
( )0 0
1 2
1 1(1), (2)( ) 2 ( ) 2 2 2
x a x af f
+ + = + =
50. ,a \ , 2 24 9 1a + = z 2 1 2 31
z a iz
+ = + , 1z ,
* ` , : ) 1 1z + = ) z ) w 3 5 2w i+ = , w . ) z w+ .
) 2 1 2 31
z a iz
+ = + ,
2 1 2 31
z a iz
+ = +
-
47
: . . 47
( ) ( )
( )( ) ( )( )
( ) ( )( )( )
2 2 2 2
2 2
2
2 1 2 3 4 9 1 11
2 12 1 1 1 2 1 11 1
2 1 1 2 1 2 1 1 1
4 2 2 1 1
3 3 3 0 0
1 1 1 1 1
1 1 1 1 1 1 1
z a az
zz z zz z
z z z z z z
zz z z zz z z
zz z z zz z z
zz z z z z z
z z z z
+ = + = + = =
++ = = + = + = + + = + + + = + + + = + + = + + + = + + + = + + = + = + =
) z 1C ( )1,0K 1p = , z 2C ( )1,0 1p = .
) ( )5 3 2w i = , z 3C ( )5, 3M 2r = .
) ( )z w w z+ = w z . ( ) ( ) ( )
( ) ( )( ) ( )
2 25 1 3 5
max 5 1 2 8
min 5 1 2 2
z w r
z w AB r
= + =+ = = + + = + + =+ = = = =
-
48
: . . 48
51. f ( ),a lim ( ) lim ( )x a x
f x f x
+ = = .
) ( )( ), ,
( ),
f x xg x
x a x
= = =
.
) ( ),a : ( ) 0f =
) lim ( ) lim ( )
lim ( ) ( ),( )
x a x ax a
g x f xg x g a g
g a
+ +
+
= = == .
lim ( ) lim ( )lim ( ) ( ),
( )x x
x
g x f xg x g g
g
= = ==
.
f ( ),a , g ( ),a g ( ),a g a
gg
[ ],a .
) ( )( ) ( ) ( ), ( ) ( ) ,( )
g ag a g g x f x x a
g
= = = =
[ ]( ) ( )
,
, , : ( ) 0 ( ) 0( ) ( )
Rolleg
g a g fg a g
= =
=
52. 23
1
xa
xa
e xA dxe
+= + , (0) 0f = .
: 2 2 20
0
3 3 3( ) ( )1 1 1
x x xx x
x x xx x
e x e x e xf x dx f x dx dxe e e
+ + += = + + + + 2 2
0 0
3 3( ) (1)1 1
x xx x
x x
e x e xf x dx dxe e
+ + = ++ +
-
49
: . . 49
2 2
0
2 2
0 0
3 3 .1 1
3 31 1
x xx
x x
x xx x
x x
e x e xH dx e e
x
e x e xdx x dxe e
+ + + +
+ + + +
( )
( ) ( )( )
2 2
22
2 2
2 2 2 2
2 2 2
2 3
3( ) 3(1) ( )1 1
1 1 33 3 31 11 11
1 3 3 1 3 3( )1 1 1
(1 3 ) 1 3 1 3 11 1
( ) 1 3 ( )(
x x
x x
x
x xx x
xx x
x x
x x x x
x x x
x x
x x
e x e xf x xe e
e xx e x e xe eee e
e ee x e x e x e xf x
e e ee x x x e
e ef x x f x x x cf
+ + = + =+ +++ + += + = +++ ++
+ + + + + = + = =+ + ++ + + + += =+ +
= + = + +30) 0 0 0 0 0c c= = + + =
, 3( )f x x x= + 3
23
( )3
1
xa
xa
f a a ae x dx a a
e
= ++ = ++
53. ,f g ,f g
( )( ) ( ) ( ) ( ) ( ) ( )f x f x g x g x f x g x + = . : ( ) ( )f x g x= .
,f g , : 0 0 0: ( ) ( ) (1) x f x g x =\
-
50
: . . 50
( )( )
( ) ( ) ( )( ) ( )
0
2 2
2 2
2 2
2 20 0 0 0
(1)2 2
0 0 0 0
( ) ( ) ( ) ( ) ( ) ( )
2 ( ) ( ) 2 ( ) ( ) 2 ( ) ( )
( ) ( ) 2 ( ) ( )
( ) ( ) 2 ( ) ( )
( ) ( ) 2 ( ) ( )
( ) ( ) 2 ( ) ( )
( ) ( ) 2 ( ) ( )
x x
f x f x g x g x f x g x
f x f x g x g x f x g x
f x g x f x g x
f x g x f x g x
f x g x f x g x c
f x g x f x g x c
g x g x g x g x c c
=
+ = + =
+ = + =
+ = + + = + + = + 0=
, 2 2 2 2( ) ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) ( ) 0f x g x f x g x f x g x f x g x+ = + = ( )2( ) ( ) 0 ( ) ( )f x g x f x g x = =
54. f [ ]0,1 ( ) ( )22 1 ( ) ( ) (1)x f x x x f x [ ]0,1x . i) (0) (1) 0f f ii) f ( )0,1 .
i) (1) 0 : (0) 0 (0) 0 (0) (1) 01: (1) 0 (1) 0
x f ff f
x f f= =
ii) ( )f x ( )0,1 , ( ) [ ]( )( ) 0 0,1 ( ) 0, 0,1if x x f x x
( ) ( ) ( ) ( )2 2 2 2: ( ) 02 2 2( ) ( )(1) ( ) ( ) 0 0 0( ) ( )f x x x f x x x f x x xx x f x x x f x
f x f x
2
( )( )
x xg xf x= [ ]0,1 2( ) 0 (2)
( )x xg xf x
=
(0) 0(0) (1)
(1) 0g
g gg
= ==
-
51
: . . 51
[ ]( ) ( ) (2)0 00,1
0,1 0,1 : ( ) 0(0) (1)
Rolleg
g x g x g g
=
=
( )f x .
55. f [ ],a . : ( ) 1 21 2 3 ( ) 2 ( ), , , : ( ) 5f fa f + = .
[ ],a 5 5
... ( [ ]. . ,f f f ).
. . .
1
1 1
1
, 25
, 2 :5
, 25
2 ( ) 2 ( )5 5( ) ( )
2 25 5
2 ( )52 ( ) 5 (1)
f a aa a
f a a
f a f a f a f af f
a a
f a f af
+ + + + + = = +
+ =
-
52
: . . 52
( )
. . .
2
2 2
2 2
2 ,5
2 , :5
2 ,5
( ) 2 ( ) 25 5( ) ( ) 5 5 2 22
5 5
( ) 2 ( ) 25 5( ) 5 3 ( ) 5
3
f aa
f a
f f a f f af f
a
f f a f f af f
+ + +
+ + = = + + + = = (2)
1 2( ) ( )(1), (2) 2 ( ) 3 ( ) 5 (3)f f af f
+ =
[ ]( ) ( )
. . .
(3)
1 2
, ( ) ( ), : ( ),
2 ( ) 3 ( ) 5 ( )
f a f f aa ff a
f f f
= + =
!!!
: ( ) 1 2 3, : 6 ( ) 1 ( ) 2 ( ) 3 ( )a f f f f = + + 6 1 2 3= + + [ ],a
6
-
53
: . . 53
56. f \ , (1) (3)f f= ( ) ( )3 3f x f x x\ (1). ( ) 0f x =
.
: ( ) ( )3( ) 3h x f x f x=
( ) ( ) ( ) ( ) ( ) ( )3 3 2 3(1) (1) (3) (1) 0
( ) 3 3 ( ) 3 3 3 (2)
h f f h
h x f x x f x x h x x f x f x
= = = =
, ( ) 0 ( ) (1)h x h x h x \
. Fermat (2)
(1) 0 (1) (3)h f f = = [ ]( ) ( )1,3
1,3 1,3 : ( ) 0(1) (3)
Rollef
f ff f
= =
57. ( ) 1 2x xf x z i z x= + \ . 1, 1 1, , 1z i z z i z < < ) .
) ( ) 0f x = . ) ( )M z .
) 1z i z < , Re( ) Im( )z a i z z= + < . ) ( )lim 1x xx z i z
) : z i a = 0 1a< < 1z = 0 1< < , ( ) 2x x ff x a D= + = \ 1 2,x x \ 1 2x x<
-
54
: . . 54
1 2
1 1 2 2 1 1 2 2
1 2
1
1 2 ( )
1 21
1 2
2 2 ( ) ( )x
x
ax x
a x x x x x x x x
x x
x x a aa a a a f x f x
x x
+ + > + >
< >0
0
f 2 \ .
) : 0 0(0) 2 0f a = + = , 0 ( )f x f 2 f 1-1, 0
( )f x .
) ( ) ( ) 11 0 1 0,1 , 11 1z i
z i K Rz
< + < = <
(1 0 ) 1z i + < ( ) 21,0 , 1R = . , , ,
.
) ( )( ) ( )( )2 21 1 1 1z i z z i z z i z i z z < < + < ( ) ( )1 1
2 2 2 2 Im( ) Re( )
zz zi zi zz z z z z i z z
yii x y x y x z z
+ + < + < + < < > >
) : , 1z i a z = = (): 1 ,x xa a < < 2 2
( ) ( ) ( )1lim 1 lim lim 1 1 0xxx x x x xx x xz i z a a
>+
= = = + = + /
-
55
: . . 55
57)i) ln 1t t 0t > . ii)
2 1( )ln
x
xf x dt
t t= . f .
iii) f . iv) 0 ( ) , 0f x x x< < > . v) ln 2 ( )f x< 1x . vi) . i). ( )( ) ln 1, 0,g t t t t= + . g ( )0,+ ( ) 1 1'( ) ln 1 ' 1 tg t t t
t t= = = .
: '( ) 0 1g t t , g :
- ( )0,1 , - [ )1,+ , 1x = , (1) 0g = . ( ) (1) ln 1 0 ln 1g t g t t t t 0t > .
ii)H 1lnt t 0t > ln 0t t ( (i)).
: 0x > 2 0x > , ( )0,fD = + .
iii).. H 1lnt t ( )0,+ ,
2 1ln
x
xdt
t t ( )( )
22 1 ln(2 ) ln'( )2 ln(2 ) ln 2 ln(2 ) ln
x xf xx x x x x x
= = . ( )0,x + :
22
( ) ln( )
2 2'( ) 0 ln(2 ) ln 0 ln( ) 0 ln( ) 0
2 2ln( ) ln1 1 2f x x
xf x x xx x
xx x
=
/
, f : - ( ]0, 2 , - [ )2,+ , 2x = (2)f .
-
56
: . . 56
iv) ln 1t t ( ) 0t > , 2 1 0ln
x
xdt
t t> ,
( ) 0 (1)f x > . 1 :
ln 1 0ln
t tt t , ,
2 2 2 2ln 1 1 10 1 0 1 ( )(2)
ln ln lnx x x x
x x x x
t t dt dt dt dt x f xt t t t t t > > > >
(1) ,(2) : ( ) 0, 0x f x x< < > . v) 1t :
ln 1 10 0ln lnt
t t t t t , ,
2 2 21 1 1 10 ln(2 ) ln ( ) ln 2 ( )
ln lnx x x
x x xdt dt dt x x f x f x
t t t t t t < < < , 0 ( )f x x 0x > .
0 0lim 0 lim 0x x
x+ + = = ,
0lim ( ) 0x
f x+ = , f .
-
-
- ..
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