simeioseis nikoletopoulos g_lykeiou
TRANSCRIPT
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. 1.
) f(x) = +
2
3 2x 4
x 6x 11x 6 ii) f(x) = 3 2 x 1 iii) f(x) = +32 log (x 2)
iv) f(x) = log x ( log4(3-x2)) v) =2
f 5x x(x) log 4 vi) f(x) =
24 x12
vii) f(x) 5 x 4= + viii) = 3 2f(x) x 1 x 2. R f(x) = ln ( x2+2x+9) = R
3. R
: )f (x)=+
24 - x(x - 1) x 1
) f (x)= 2x - 2 - 1
+ 34 - x - x
) f (x) =2x - x
x - 2 - 1 + 1
3x - 8 - x
) f (x) = 5x - 3 - 1
) f (x) = log (x2 + x - 2) + log +x 33 - x
) f (x) = xe - 1 + 1 - lnx ) f (x) = x2x - 1
+ 1x - 1
, x [0, 2]
. 4. C f , : I) f(x) = x2-5x+6 ii) 3xf(x) e 1= iii) 5f(x) log (x 3) 1= iv) f(x) x 3 2= 5. f (x) = x2 - 3x + 2. ) f (1), f (0), f (-3), f (2) ) Cf ) f (t), f (xt), f (x + h), x, t, h R.
6. f (x) = x 1
lnx . ) f. ) f (x) = e x . )
7. ( ) ln( 1) f x k x= + + 2 1e 2 . : i) , R
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ii) fc 3 .
8. , :f g R R , : = + 2( ) ( ) 4f x g x x x R . fc gc .
9. , :f g R R : + + = 2 2( ) ( ) 1 2( ( ) ( ))f x g x f x g x , x R
10. x R f g
( ) ( )( ) ( )( ) ( ) ( ) ( )
2 2
2
3 2
x +2x+3 -x -x-2
-x-1 2x
i) f x = x +2x+1 g x = x +2x+1
ii) f x = e g x = e
v) f x = ln e +1 g x = ln e +1
11. :f R R : 2 2( 2) f(3 x) 0f x + + = , x R . f .
12. ) + += =
2
2 3x 2 x 2xf(x) , g(x)x 1 x x
) + += =
x 3 x 3f(x) , g(x)x 2x 2
) = + = + f(x) x 4 x 5 , g(x) x 5 4 x
13. f (x) = x + 1. ) f.
f1 (x) = 2x - 1
x - 1 f2(x)=
++
3
2x 1
x - x 1 f3 (x) = ( )2+x 1
f4 (x) = x (x 1 + 1) f5 (x) = lnex+1 f6 (x) = eln (x+1)
) R .
15. . f (x) = +x 1x - 1
, g(x)=2+ +2
22x 2x
(x - 1),R, x > 0.
) f, g ) f = g; . f=g .
.
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f g , : =( ) ( )f x g x .
i) =
2
21( ) xf x
x x = + 1( ) 1g x
x
ii) =
2( ) ln
1xf x
x = ( ) 2ln ln(1 )g x x x
16.
+ += =
2
2x 3x 1, x [0,2] , x ( 2,1)3x 6f(x) g(x)2x 3 , x (2,6) x 1 x [1,13]
2f+3g, f.g , f 2g
17. f (x) =x,
+
>
2x 1, x 2 x 2
g (x) = ,
<
0, 0
g(x)1, 0
= >
: ) f g !!!!!!!!!!!!!!
.
19. = +f(x) x 3 = g(x) ln(x 2) . f g . g f . f f v. g g 20. f g, g f, f f, g g
21. f i) = + 2 ,(f g)(x) x x 3 g(x)=x-1
.
+ > += =+ +
2x 3, x 0x 1, x [1, )f(x) , g(x)2X 3. x ( ,1) 3x 1 x, x 0
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ii) = + + 2 4 ,(f g)(x) 3 x x g(x) = x2 iii) = + = (g f)(x) 5x 4, g(x) 7x 6
22. f [0, 2].
: ) f (x2) ) f (x - 4) ) f (lnx) ) 2f( 1 x ), ) f(3 x 2) ) f( 3 x)
x 3+
23. [ ]: 2,1f R
: = ( ) (2 3)g x xf , 2( ) ( )h x f x= ( ) (ln )q x f x=
24. : (0,1] Rf . ( ) ( 2) (ln )g x f x f x= +
25. 2( )2
xf xx
=
( ) 1g x x= .
: ,f g g f 1ff
26. x R + =3 3f(x) x ( ) =f f (x) x
27. f ,
+ = f(x) 12f 3, x Rx x
f.
28. f(x)
( ) ( ) ( )( ) ( )
( )
2 3 6 3
2 x 2
I) f 2x -1 = x -3x+2 II) f x = x -2x +1 x >0
III) f lnx = x - x+2 IV) f e -1 = x -3x+1
V) f(ln2x) = x+3 x >0
29. = 2 1( ( )) xf f x x R
: ) (2 1) 2 ( ) 1f x f x = , x R ) ( ) 1f x = .
30. f : R R, + +f( ) f() f() , . ) .f(x) 0 ) )f() f() f( ,R.
31. ( ) = + x ef g (x) e x = > .g(x) lnx 1, x 0 f 32. = f(x) g(x) x = 2g(x) x f(x), x R,
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= (f g)(x) (g f )(x) xf(x) 33. f : R R = + 2f(x 1) 2f(3 x) x 1, x R. ) = + +2f(x) 2f(2 x) x 2x 2 ) = +2f(2 x) 2f(x) x 6x 10. ) f
34. f 2f (x) - 3f ( 1x
) = x2, x0, f (2).
35. = +2f(f(x)) x x 1 .f(1) 36. = f(f(x)) 3x 2 = f(3x 2) 3f(x) 2, x R
37. : 3 1, 1( )2 , 1x x
f xx x
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) f gc c .
) ( )( ) , 0( )
f xh x xg x
= > hc
,x >0 .
44. f : R R (0) 0f =
=
( )( ) ,1x
f xg x x Re
. ( ) 0g x < 0x .
45. (1,5)A (5,-2) . : ) f . ) f f ( (e )) 2xf f < .
46. f : R R :
+ = + >( ln ) ( 1) ln 3, 0f x x f x x x . f , ( ) 2f x = ( 1) 2xf e < .
47. f : R R f ( ,0] f . :
) + = +( ) (5 ) (3 ) (7 )f x f x f x f x ) + = +5 3 7( ) ( ) ( ) ( )x x x xf e f e f e f e
48. f :R R . :
( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( )2
. f x > f 3 . f 2x+1 0 . f f 3x -1 < f f 2x+5
49. ( ) 1-x 1f x = e + -2 , x >0x
.
. f ( )0,+ . ( )0,+ : 1-xxe -2x +1>0 50. . ( ) ( )xf x = e +ln x+1 -1 . . . . : ( )2x 2e +ln x +1 >1 51. [ ) f : 0,+ R ( ) ( )2f x =3x+ln x +1 x>0 . . .
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. : ( ) ( )
2
2
4
x +1 +1ln > 3 x - x - 2
x +1
( )2, + 52. : 1lnx < -1
x
53. ( )g : 0, R+ (1,-2) , (2,-3) ( ) ( )f x = lnx - g x x > 0.
I. g . . . . ( )22lnx
+
g(x) 0> > 0.
.
59. , : [0,2)f g R
= + +( ) ( ) 2 f x g x [0,2) fc gc .
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60. , :f g R R + =2 2( ) ( ) 1f x g x
x R . fc gc 1=
=( ) 2 ( ) ( )h x f x g x 61. ) .
, . ) .
. -
62. . :
1. 2. 3. , .
63. :f R R . , f
,x R : ) 1 ( ) ( )( )( ) ( )
f x ff xf x f
++ =
+ ) ( ) ( ) ( )f x f x f + = +
64. f : R R
f (x + y) + f (x - y) = 2f (x) + f (y) x, y R. ) f . ) f . ) x R f ( x ) = f (x).
. 1-1
65. 1 1 :
1) 3f(x)= x -2 2) x+1f(x)= e -4 3) x -1f(x)=x -2
4)
2x -3 ,x 1f(x)=
lnx -1,x >1
5 )
2f(x)= x -6x+10, x >3
66. 1 1 :
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( ) ( ) ( ) ( )x
3x
2e -1) f x = 2x -4 ) f(x)= ) f x = 2- 4+3x ) f x = 3ln x+2 -1e +1
67. 1-1 ( ) ( ) ( )2 4 2i)f x = x -3 , ii) f x =5+ x -4 , ii) f x = x -5x +7 68. f ,g : R R :
f(x)>0 R g 1-1 15 3x-1f(f(x))= lnf(x)+g (e ) 1 1 .
69. . 1 1 1 1 .
70. f : R R : f (f (x)) x=
x f(x)g(x)= e +e , 1 1 . f 1 1 . g(f(x))= f(x) .
71. 2f(f(x))= x - x +1, x R . ) f(1)=1 ) g(x)= x 2 x + f(x)+1 , g 1-1
70. : R f R
( )( ) *, x += + f f x ax R : ) f 1-1 ) f R
) ( ) ( ) , x = f ax af x R ) ( ) , x
+=
f xf x R
a
71. : R f R : ( ) 5( ) ,f xf x e x+ = x R . :
f 1-1. H f .
72. f , g 1-1 g f 1-1
73. f: R g : f(A) R. g f
1 -1, : i) f 1 - 1 ii) g 1-1
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.
1-1 ( ):f A f A ( )1 :f f A A x A ( )y f A ( )f x y= ( )1 f y x =
1-1 ( f "1-1"f )
1 ffD R = 1 ffR D =
f , 1-1.
y x=
y x= , f .
( )
.. ( ) 1f xx
= ( ) 4f x x= . ( ) ( )1f x f x= . : ( )( )1f f y y = ( )( )1f f x x = ( )( ) ( )1 1f f x f x = :
1. 1-1
2. ( )f x y= f x 3. x f(A) f 1f 4. f 1f (x) 1-1
f -1(y)=4-y
y=3 x=f -1(3)=1
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.
74. ( ) xf x = e +1 . f 1-1 N f. . . f f-1
75. ( )x
x
e -1f x =e +1
. f . . f f-1
76. ( ) ( )xf x = 2+ln e +1 . f . . N f. . f f-1
77. 1( ) ln1xf x x
= +
) f . ) 1f . ) f 1f . ) : f 1-1 1f
.
78. . f 3x 2x(x)= 2- ln(3+2e ), f(x)= 3- 3-e ,
2015f(x)= 2x +5
79. f :[3, ) R+ 2f (x) x 6x 10= +
1. 2.
3. g :[1, ) R+ g(f (x)) x= x 2
80. : R f R ( ) 0f x > ln( ( )) ( ) xf x f x e=
x R . : 1. ( )f x >1 R
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2. f 3. f 81. : R f R : 3f(x)-2f (x)+x -1= 0 x ) f ) 1f .
82. f(x+y)=f(x)+f(y) x,y R.
. f 0 f 1-1 .
. f -1(a+b)=f -1(a)+f -1(b) a,b R.
( f(R)=R)
( f(x-y))
83. f(xy)=f(x)+f(y) x R*+ . . f =1 *.
. f 1 1-1 .
. f 1-1 a,b f -1(a+b)=f -1(a)f -1(b).
f 1f : ) f ( )f x x= ( )1f x x = ( ) ( )1f x f x=
) f ( ) ( )1 y f x y f x= = 1f fx D D
84. :f ( )f R R= .
1. f 1f .
2. (0) 1f = 1f . 3. (2) 3 (1) 2f f= = , :
1 2(1 ( )) 1f f x x + + < .
85. A : A(3,2) B(5,9) f, : : 1 2f ( 2 f ( x 8x )) 2 + < f
: 1 2f ( 2 f ( x x )) 9+ + =
-1 -1f(x)= x, f (x)= x , f(x)= f (x)
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86. ( ) 3 3, xxf x e x= + f ( )1 0f ( ) ( )1f x f x= ( )( )1 1 1f f x x + <
87. () =3+-1 .
= . ()= -1() . -1(3+2)>1
88. f : (3,2) (5,9). ) f ) -1 ) f(2+-1(2+))=9 ) f(-1(x2-8x)-2)
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) -1f y = x f R . 93. ( ) ( )f x = ln x -1 + x - 2 ) . ) ) -1f f f R . 94. ( ) 3f x x= . 95. ( ) 5f x = ln x+3x -2, x >0 f . fC 1fC
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15
.
1. f . : )
+x 2lim
f (x) )
-x -1lim
f (x)
) 1
limx - +
f (x) ) 1
lim-x f (x)
) 1
limx +
f (x) ) 2
limx
f (x)
) 3
limx
f (x)
y
23
4
1
-2
-2
1-1 2 3 x
B. 0x x
f(x)lim
g(x) 0g(x ) 0
2. : )
++2x 2
3x 5limx 1
)
+ 45x 1lim ( (x 3) ) )
+
2x 3x 2 5
limx 6
v)
+ + + +
+
3
x 1
x 2 x 2x 1lim
x 3 1 v)
4 3 4lim 3 11
x xxx+
.
3. : i)
2
2x 1
x -1limx - 3x +2
ii) 1x
lim
31 3
1 x 1 x
iii)
2x 2
(x 2) 3 xlimx 4
iv)
+ +
2
3 3x x ( 1)x lim
x
4. 3 3
2 4x 1 x 1
x -1 x -3x+2lim ) lim2x +3x -5 x -4x+3
)
.
00
-
9
16
.
5. : i)
+
2
x 2x 4lim
3x 2 5x 2 ii)
+ +x 11 3x 2lim
3 5x 4 iii)
+ 3 3
x 01 x 1 xlim
x
6. : i)
3
4x 1x 1limx 1
ii)
+ +
3
x 17 x 3 xlim
x 1
iii)
x 3
f(x) 2 f(x)limf(x) 1
=x 3lim f(x) 1 iv)
1limx
2
2
x - 1 + x - 1x - 1
.
7. 2
2 3x 2 x 1
x +7 -3 x -1- 2x - x -1lim limx -4 x -1
a)
)
.
8. : )
2x 2
x 1 x 3lim
x 4 )
+
+ +
3
3x 1
x 3x 1 xlim
x 5x 4 2
)
+
2 2
x 3
x 9 x 6x 9limx 1 2
v)
+ =
2
2x 2 x 2,
f (x) 4 3 f(x) 5lim , lim f(x) 1
f (x) 1
9. : ) 3 3 1
lim 1 21
+
+
x xxx
)
+ +
+
2 2
x 1
x 1 x x 2lim
x 2 1
) 5 35 1 3 1
lim0
+ +
x x x xxx
.
10.
<
+= +
2
2
x 1, x 1x 1f(x) , x [ 1,1) (1,3)x 13x 1, x [3, )x 7
) lim f(x)
x 2 )
x 1lim f(x) )
lim f(x)
x 2 v)
lim f(x)
x 3
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9
17
11. x 1limf (x)
( )
2
2
2
x -1 1
x -1
)
12.
( ) ((
2
2
x + x + x 0,2
f x = x +1 x 2,3
2x - 3 x 3,4
=2 =3
13. 2 2 2x f (x) 4f(x) 2x 5x 2 x 2
lim f(x)
14.
2x 0
5f(x) 3x 6 x , x R lim f(x) 15. f : R R + + +3 2x 1 f(x) x x x ,
i)
x 1limf(x) )
x 1
f(x) 1limx 1
)
+
2
x 1
f (x) 1limx 3 2
16. + + + 3 2x 1 f(x) kx x x, x R f
(1,1)
x 1
f(x) f(1)limx 1
17. + 23 x 5 9 (2x 4)f(x) 4(x 2x)
x 2 x 2 3
f(x) 1lim f(x), limf(x) 1
18. f : R R x f(f(x)) f(x), x R
+
+ +
1
2x 1
(f f ) (x) ( 1)f(x) lim , N(x 1)
19. x 12
limf(x)= 2
f(x)-g(x) x -1
( )
x 1limg x
20. ) 2x lim f (x)= 0
, :
x lim f(x)= 0
) ( ) ( )( )2 2
xlim f x +g x = 0
( ) ( )x xlimf x ,limg x
.
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18
( )
21. ( )( )
( ) ( )
x
x
limf x = 0
limg x = 0
f x >0,g x >0
( )x 1limg x
.
22.
( )
x 0 x 0 x 0 x 0
2x 0 x 0
-15 2 3 i) lim ii) lim iii) lim iv) lim -1 2 3v) lim vi) lim
+5x 3+4 -2
23. )
( ++2x 0
x) (x)limx x
) 0
limx 2
2 1 - 1 - x x
)
+ 2x 0
x 25 5lim x
v)
4 4
x 4
x xlimx 1
v)
+ x 0
1 x 1 xlimx
24. : 0
limx
x + 2x + ... + xx
= 28.
25. : 0
limx
x 2x ... xx
= 120
26. )
x 0
1lim x x
)
x 0
1lim x.x
27. 3 3 3 3f (x) 8x x x x 0
f(x)limx
28. 2 3 3xf(x) x x. , x Rx
. x 0
lim f(x)
29.
+ = , , =2 2x 0 x 0
xf(3x) f( x).(ax)f(x)lim 3 R lim 7x 4x x
30. f(x) 2 1x , x Rx
=x 0lim f(x) a , .
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9
19
. ..
31. )
+ = 2x 2lim (f(x) x x 5) 7,
x 2lim f(x)
)
= = +2x 1 x 1g(x)lim(x 1)f(x) 5 lim 4,
x 3x 2
x 1lim f(x)g(x)
32. )
= + = ,
x 2 x 2 x 2 x 2lim[f(x) 2g(x)] 3 lim[2f(x) g(x)] 6 lim f(x), lim g(x)
)
( )( )
( ) ( )x 0
3x 0
f xlim =1
6x
lim 1+x -1 g x =10
( )( ) ( ) ( )x 0 x 0
f xlim ,limf x g x
g x
)
( ) ( )
( ) ( )x 0
x 0
f x 3x-xg xlim =5
4xf x 3x+xg x
lim =74x
( ) ( )x 0 x 0limf x ,limg x
33.
= =
+ + x 0 x 0g(x)f(x)lim 1 lim 5
1 x 1 x 3x 4 2
x 0
f(x)limg(x)
34. f(x)+f(x-)=0 , a
= ,
x x 00. lim f(x) 2 lim f(x)
35. f R , f(+) = f( )+f() , R
=x 0
f(x)lim 4x
x
f(x) f()lim , Rx
36.
+ =2
22x 0
f (x)lim g (x) 0x
,
= =x 0 x 0lim f(x) lim g(x) 0
37. ()= + + 3 2x x x, 0
= =x 0 x 0
f(x)lim 1 lim f(x) 0x
x 0
(f f)(x)limx
38. + 2 2f (x) 2f(x) x 0 R ,
=
x 0lim f(x) 1
.
39. + + =
2x 2ax 5f(x)x 3
, R
=x 3lim f(x) 12
40. ,
+ =x 1
x x 2lim 5x 1
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9
20
41.
+ + + =
3
2x 1
kx ( 2)x 4lim L R(x 1)
, ,, L.R.
42. ( )3 2
2
x + 8x + 3x - 2f x =x -
.
f (0,1/2 ) ( )x 2limf x =
,
43. ( )2 2
1x2
2x + 5+2 x+-1lim =151x -
2
, ,R
44. f R x 0
f (xlim )x
=2 .
: R x 0
(lim 42
2
2x)f(x)+x x-3xf(x)
=
. 0
45.
) ( )
+
2x 22x 3limx 2
)
x 1
3x 2limx 1
)
x 0
2 1limx x
46.
) f (x) = x - 2(x - 1) (x + 2)
x0 = - 2 ) f (x) = 3 2x + xx + 1
x0 = - 1
47. i) 2 2x 22 xlim -
x - 4 x -2x
ii) x 0
x -1limx
48.
0
.
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9
21
) 2
x 0
cx +2x +dlimx
) 2 2x x - - x -
limx -
) 2
2x 2
x +(b+2)x -16limx - 4
iv) 2 2 2
x 0
c - c - xlim , c 0x
49. , R
+ + + =
2
x 1
x x 2lim x 4x 1
50. , , R ( )
+ + + + =
3
2x 2
x ( 1)x 2lim 6x 2
. ..
51.
+=
x 2
1 xlim
f(x)
x 2lim f(x)
52.
= ++
2
2x 1
f(x) (x)lim(x )
x 1lim f(x)
53. f R ( )
x 3lim x 3 f(x) 2
= m
2 3 2
3 2x 3
(m 1)f (x) f (x) 3lim 2.mf (x) f (x) 2
+ +=
+ +
54. f R
= +x
f(x)limx
+ + 2
x 0
1f(x) bf(bx) f(x )xlim
x,
55. f g R >f(x) 0 >g(x) 0 x R .
+ =
0x xlim (f(x) g(x)) 1
= +0x x
f(x)limg(x)
0x xlim f(x)
0x xlim g(x).
56. = 1 2g (e )
= xg(x) e . pC (0,0)
= x 0
P(x)lim 1x
=x 1
,P(x)lim 1x 1
)
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22
)
+ + x 0 2
P(x) x 2limx 1 1
x
y
0
y
x 1
1
58. f. 1. f .
2. , ,
.
57. f . ) :
1lim
-x f (x),
1lim
x +f (x),
limx +
f (x), limx
f (x).
)
1
limx (x) f
1 ;
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23
) ( )x 1limf x ) ( )x 3limf x ) ( )x 5limf x ) ( )x 7limf x ) ( )x 9limf x . 4 . 7 B3. , , .
) ( )x 21lim
f x )
( )x 61lim
f x ) ( )( )x 8limf f x
9 B4. f . .
3 B5. 0x f
( )0f x 0 = . 2 59. + .
) f(x)= - 3x 4 + 5 x 2 +1 ) ( )4 3x +lim 3x - x + 2
) ( )3 2x -lim 2x + 4x + 3x +1
60. + - . ) 3 2f (x) (2 1) x ( 1) x 3x 2 = + + f(x)=( 2 3 22) x 2x 6x 7 +
61. 3 2 3
3 5x + x -
4x + x - 2 3x - 5xlim ) lim3x + 5 2x +1
)
62. + - .
i) 2
2
x x 5f (x)
x x 3
=+
ii) 2
2
x 2x 3x 1f (x)
4x 1 x
+ +=
+
63. ( )( )
3
2x -
-2 x +2x -1lim
-1 x +3
64. ( )3 2
2
x -4x +3f x = -x+x +4
( )lim f x = 2x -
65. R . )
5 4
4 3
(2+4)x +2x -5f(x)=(+3)x -7x +3
.
III.
-
9
24
) 22x +6x -2f(x)= ( +x -1)3x -2
) 2 3 2
2
( --2)x -(-1)x +3x -9f(x)=(-1)x +3x -7
.
66.
( ) ( )2 2 2x + x -lim 4x + 5x - 3 - x +1 , ) lim x + 4x + 3x)
67.
) 2 55
7 67x +
x +3x -5 + x -4x+2limx -8x +2 +6x -3
) 2
63x -
4x +2x+6 +7-2xlimx -4x+9 +x -7
68. ) 2 2
x +lim( x +1- x -1)
) 2xlim [x( x 1 x)
+ iii) 2xlim ( x 2x 5 x 2)+
+ + +
iv) 2xlim ( 9x 5x 7 3x 2)
+ + v) 2 2
x
x 2x 5 x 4x 2limx 2 x 7+
+ ++ +
69.
) 2 2xlim ( x 4x 5 16x 5x 3 5x 2+
+ + + + + ) 4 2 4 3xlim ( x 6x 1 4x 4 9x 2x 2)
+ + + +
) 2f (x) ( x 4x 1 2 x)= + + iv) 2 2f (x) ( x 3x 1 4x 3 x)= + + + 70. 3 2 24x 5x 3 f (x) 2x 3 4x x 6 + + ,
xlim f (x)+
71. , R 2
xlim ( x 2x 3 2ax ) 3+
+ + + =
72. > 0 4 4 2(3 x )f (x) 3x x 1+ + ,
xlim f (x) 3+
=
73. x
xf (x) 2x 3f : (0, ) R, lim 6x 4
m+
+ + =+
xlim f (x)+
74. x x
f (x)lim 2, lim (2x x)g(x) 5.x x+ +
= =+
xlim [f (x)g(x)]+
IV. O
75. ) x
1 1lim ( )x x
m+
) 6x2lim (x )x
m+
-
9
25
)x
xlimx
m
v) 3 2
x 3 2
15x 2x 6xlim 64x 5x 12x
m
m +
+ +
76. 2x
4lim ( 16 x 4)x
m
+
77. ( )x + x +
) lim ) lim 2x+
78. 2
2 2x - x +
2 +) lim ) lim -
79. 2
x + x - x +
1 1 1) lim , ) lim x . , ) lim x.
80. :
) 2x.xlim 2x + x +1 )
13 2x -2x +1 +x+2x
lim 2x + x - x+2006
V.
81. )
x x
x 2 x 1x
2 4 3 2lim3 2
++
) x 2 x
x 2 x 3x
4 3lim6 4
+
+
++
, > 0
) x 2 x
x xxlim
+
+
v) 13lim
1 3 ++ +
x
xx v)
1
14 3lim4 3
+
+
x x
x xx
82. f (x) = ln 2x
3 -x , : ) f
) + x
lim f (x), x
lim f (x), 3 x
lim
f (x), 0 x
lim
f (x).
83. f (x) = n
+x
x 22 , > 0. )
f. ) 0 x
lim
f (x), + x
lim f (x). )
f (x) lnx > 0 + x
lim (f (x) - nx).
-
9
26
84. 12% x
f (x) = 16x 362x 3
++
. ) :
f (x) = 8 + 212x 3+
. )
8%. ) , ; 85. () = + -1-1 + -2-2 + + 1 + 0
Q() = + -1-1 + -2-2 + + 1 + 0
( ) ( ) ( )H x Q x = , i)
(,+ ). : -1 -1x
-lim H(x)=+
.
86. f(x) f(0) = -4,
2
( ) ( )lim 2, lim 3, , 0 +
= = + +x x
f x f xx x
.
f(x) ,. (University of New York).
87. ) f: 2 2 2 2f(t)= t +(x + y )t+2 - t +4yt+3 , (,y) .
lim ( ) 0,
=t
f t .
) f: 2 2 2 2f(t)= t +2xt+3x - t -2yt+5y ,
(,y) .
lim ( ) 2,
= t
f t . )
.
88. () limx 2P() = 3
-2+1
limx x02
P() -2+1
=3, 0.
89. f: : 3 3f (x)+f(x)= x x R
) f 1-1. ii) f(x) = 0.
iii) 0>x ( )
-
9
27
90. g 2x +ax +g(x)= , R
x -1
1
lim ( ) 2
=x
g x f:
3 2 2 20
( )lim f ( ) 2 ( ) 4 ( ) 8 0l m
= + =x
f x x xf x x f x x xx
x.
) , . ) , , .
91. R)0,(:f xxxxx5x2x)x(f 2247 +m+++= (1) () )x(flim
x () 2x x
)x(flim
92. RR:g,f : 32xx)x(flim 2
2
x=
+
+
2x
)x(glim 2x =+ . R 6x)x(g)x(f
x3)x(g)x(f)1(lim
2
x=
m+++++
+
.
-
9
28
-
9
29
-
- 9
28
.
I. 0
1. 0=0 :
i) f(x)= ( )1 , x 0
1 , x 0
x xx
= ii)f(x)=
1-2 , x 0
2 , x 0
+
=
iii) f(x)= 1 1
, x 0
-2 , x 0
+
= iv) f(x)=
9 5 , x 0
4 , x 0
=
2. ( )
2 1x ,x 0f x = x
1 ,x = 0 : 0=0
3. , f f(x)= 2 , x
-1 , x
+
= =
< 0 , f(a) 0 ,
0
0
0
00 xa
)x(fx
)xa(f:)a,0(x
=
( )
39. f [0,1] 0 f(x) 1 [ ]0,1 , , [0,1), :f 2005() + = f() 40. f, g [ ], = ( ) ( ) [ ]f g , = = [ ]0x , , ( )( )0 0g f x x= . 41. f = [, ]. : f (+-)
. , , : f (+-) = f(). 42. f = [, ] >0. ,
, : f() = f ( ) f ( )
1 + +
.
42. f [0,4] f(0)= f(4), , [0,4]
- = 2 , f()= f(). 43. f [ ], 1 + : ( ) ( )f f 1 0, . + + = ( )f x 0= [ ], 1 + 44. f: f(1) + f(2) + f(3) = 0.
f() = 0 .
45. : y x 1 = + ( )f x 2 2x= ( )0x 1,0 .
.
. BOLZANO
-
- 9
34
46. f(x) = x2 + x + g(x) = - x2 + x + 0.
1 f 2 g 1
-
- 9
35
i) 2000 20041 2 0
1 1 + +
+ = +
(-1,1).
ii) 06 4
+ =
( , )6 4 .
56. f(x) = ex- -1 g(x) = ln(x+1) + , 3(0, ).4
i) f() = [,+1].
ii) f() = [,+1],
: xlim g(x) .
=
57. f,g R f(1)+f(2)++f(2014)= g(1)+g(2)++g(2014)., { }1,2,3,..., 2014 f() =g()
58. f ,g [0 ,] . f fggf = [0,] f( )=g( )=
59. 2 2 2
0, ,, 0x x 1 x 1
+ + = +
) (-1,1).
) 1 2, 2 2
21 2
1 1
+ = .
60. f ( ) 3 2f x x x x = + + + : 0 >
0 + + + = 3 2 0 + + > ( )0x 0,1 ( )0f x 0= 61. f : : ( ) ( )f xf x e 5 4x+ = x ( )f 1 0= : ) f . ) ( )( ) ( )3f f x f 5 10x 0 = ( )0,1 62. f(x)=ax3+bx2+cx+d , a>0 , d
-
- 9
36
63. ax5+bx4+cx3+dx2+ex+f=0 f>0, a+b+c+d+e+f=0,
5a+4b+3c+2d+ e >0 (0,1)
64. f : f3(x)+f2(x)+f(x)=x3-2x2+6x-1,
x R , , R 20 , n N * : xn= .
( ;)
68. f, :
63 x2x+8 - x+4 xf(x) +x6
, x 4
f(0) (0, 1] ,
f() = 6
+ 6 .
69. f , :
2
x 1
f (x)lim 4 4(x-2) (x-2)f(x) x 4 x .x 1
=
Cf
= x2 - x + 1
(1,2).
70. . f 11 . , ,
< < , : f() < f() < f() , f() < f() < f() .
-
- 9
37
. f 11 ,
.
( ).
.
f R f(f(x))=-x
. A f R f(a)+f(b)=f(c)+f(d) a,b,c,d
f
71. f R f(x).f(f(x))=1 , f(9)=1/9 f(1) 72. f R ( )f 8 7= x
: ( ) ( )( )f x f f x 1 = ( )f 4 .
73. [ ]f : 2,10 , ( )f 2 10= ( ) ( )( )f x f f x 5 = ( )f 10 ( )f 5 . 74. [ ]f : 1,2 . ) ( ) ( ) [ ]x 1g x f x e , x 1,2= + . ) [ ]0x 1,2 ( ) ( ) [ ]0x0f x f x e x 1,2< + . 75. [ ]f : 2,10
[ ]0x 2,10 : ( )( ) ( ) ( )
0
3f 3 5f 6 2f 8f x
10+ +
=
76. f [ ,] , 1 , 2,, [,] 1 , 2,, R+
(,)
kkk
xfkxfkxfkf+++
+++=
...)(...)()(
)(21
2211
77. f [0,1] f(0) = 2 f(1) = 4. f , (0,1)
f(): =
1 2 3 4f f f f5 5 5 5
4
+ + +
II. .. -...
-
- 9
38
78. f: [ ], . ,
32-4000+2001 = 0, [,]
: ( )2 2f f f 2000f3 2 2 3
+ + + + + + =
.
.
78. f f 2(x) + 2= 5
= (0,5). f 1.
2. . 3. f , f(1) = -2.
79. f : f 2(x) = 2exf(x),
x
80. f : :
(f (x) 1) (f (x) 3) = 0, x .
81. f : :
f 2(x) 2(f (x)x = 1, x . 82. H f R ,f(0)=-2 2f ( x ) xf ( x ) 4 0 , x R+ =
f .
83. f : ( ) ( )2f x 1 2f x x= + x .
84. f,g: [0, + ) g(0) = 1
x[f(x)-g(x)] = x [ 3f(x)+g(x)]. ) f(0). ) [0,4] f(x)0, :
1. (x-2)f(x) +x f(x+2) = x(x-2) (0,2).
2. g(x) > 0 [0,4].
-
- 9
39
85. f [1, 4] , f(x) 0, x[1, 4], f(1) > 0 f(1)f(2) = f(3)f(4) . :
. f(x) > 0 , x[1, 4] .
. g(x) = f 2 (x) f(1)f(2) [1, 2] .
. f .
86. f,g : R R f(x)g(x)=ex , x R
f(1)>3 g(2)>2. N : ) f(x)>0, x R
) x0 (1,2) g(x0)=x0
87. f R , f(0)=5 2)x(f Rx
f .
88. H f R , f(0)=-1 f
.
89. [ ]f : 3,3 ( )3 4x f x 27+ = [ ]x 3,3
f ( )3,3 . 90. f, 4x 2 + 9f 2 (x) = 36 , x(3, 3) .
f(0) = 2 .
91. ( ]f : 0,1 ( ) 1f x lnxx
=
) f. ) f. ) ( ]0x 0,1 : 0 0 02x lnx 2 3x= 92. f(x)= 4- x - 2+x . . f . . f. . f(x) < 0 .
IV.
-
- 9
40
93. f (0, +)
+x 0lim f(x)= R
x +lim f(x)=R,
x0 > 0 , 0x +10 0f(x )+e +ln(x )=1 .
1. f : ( ) += +2x 1f x 3e 5x 3 . ) f. ) f. ) ( ) =f x 0 R.
2. f : ( ) 20112 5 7,f x x x x R= + . i. f R . ii. ( ) =f x 0 . iii. f .
3. f ( ) = +xf x 4 e 2 3 . i. . ii. . iii. 1f .
4. f ( ) ( )= + +f x 2ln x 1 1 3 . i. f. ii. f 1-1. iii. 1f . iv. ( ) + =1f 1 x 2.
5. f ( ) = +
x1f x 3x 22
.
i. f. ii. x R 2011. iii. : +
-
- 9
41
7. ( )x 1
limf x , :
i. ( )
= +x 1
2x 1limf x
ii. ( )
= +x 1
f xlim
4x 3 iii. ( )( )
+ = + x 1lim f x 3x 4
8. f : 1,5
( )A 1,8 ( )B 5,12 . i. f .
ii. f 293
.
iii. ( )0x 1,5 : ( )( ) ( ) ( )+ +
=02f 2 3f 3 4f 4
f x9
9. f ( ) ( )= + xf x ln 3e 1 2.
i. f. ii. f . iii. 1f . iv. ( ) ( )< 1f x f ln5 2 2.
10. f ( ) = 3f x 2x 3x 1. i. f. ii. f . iii. ( ) =1f x 2 iv. ( ) 1f x x 1
11. : Rf R : ( )( ) ( )+ = +f f x f x 3x 2 x R .
i. ( )1f 1 . ii. ( )f 3 . iii. ( ) =1f x 3
iv. ( )( ) ( )
+ ++ x
3 x x xlimf f x f x 2
.
12. R f :
( ) ( )
+ =
2x 1f x x x 1
lim 2x 1
i. f ( )M 1,1 .
ii. ( )
2x 13f x 2 1
limx 1
.
13. f ( ) += +
x 1f x 2ln 31 x
.
i. f. ii. f .
-
- 9
42
iii. f 1f . iv. : ( )
x 1limf x ( )
x 1lim f x .
14. : Rf R g ( ) +=
x 2g x ln2 x
.
i. f g . ii. h : ( )( ) =h g x x . iii. h .
15. R f g : ( ) f x 0 x R . ( )A 2, 1 . = 1 1 =2 5 ( ) =g x 0 . : ) f R . ) ( ) 0
: + =
4 1 3 1ln2 2
17. : Rf R : ( ) ( ) = 32f x 3 2x 3f x , x R . i. R. ii. f R , f 1f . iii. ( ) =f x 0 . iv. f 1f .
( ) ( )f +1 2- f + =0-2 +1
18. f 3,3 ( )+ =2 23x 4f x 27
x 3,3 . i. ( ) =f x 0 . ii. f ( )3,3 .
-
- 9
43
iii. f.
iv. ( ) =f 1 6 ( )
x 0
3 3f x2lim
x.
19. [ ): 0,f R+ :
( )+ + + + +2 8 2 xx 2x 9 3 xf x x 3x 3
>x 0 . :
i. :
+ + 2
x 0
x 2x 9 3lim2x
. ii. :
7
x 0
2lim xx
. iii. : ( )x 0
limf x .
iv. ( )f 0 .
20. : Rf R : ( )( ) ( )+ = +f f x 2f x 2x 1 x R ( ) =f 2 5 . i. ( )f 5 . ii. f . iii. ( )1f 2 . iv. : ( )( ) + =1 2f f 2x 7x 1 2 .
21. : Rf R ( ) = + +2 2x xf x 2 1 ,x R R . i. f R. ii. ( ) = f 0 2 f.
iii. : ( )+
+
x
x xx
2f x 3lim , 3
3 2 4 3.
22. : Rf R : ( )+ +4 4x 1 4f x x 2
x . i. : ( ) 1 1f 04 2
( ) 1 3f 1
2 4 .
ii. :
4
x 0
1lim x fx
. iii. :
+
+
5
2x 0
1x f 4 3xxlim
2x 3 x.
iv. 0,1 , ( ) =f 0 .
23. i. ( )
=
x 0
2f x 4lim 2
x ( )
x 0limf x .
ii. : Rg R : ( ) + +xg x 2 2 x x x x . ( )
x 0limg x ,
. iii. : ( ) ( )( )
+
+
2 2 2
2 2x 0
x f x 2xlim
x x g x
24. : Rf R : ( ) ( )+ = +33f x 2f x 4x 1
x R .
-
- 9
44
i. f 1f . ii. 1f . iii. f 1f , =y x . iv. : ( ) ( ) = x 1f 2e f 3 x .
25. ( ) = + f x x 1 1 ( ) = g x 2 x . i. f g. ii. f g . iii. f 1f . iv. f f g .
26. f ( )
+
2
2
2x x , x 0x x
f x ,x 0
8x x 16 3x, x 0
i. , . ii. : ( )
+xlim f x .
iii. : ( )xlim f x .
iv. ( ) ( )= +f x 2ln 8x 1 ( )0,1 .
27. f ( ) ( )( ) ( )
( ) ( )
+
=
+ +
2
3 2
2
x 5x 6 , x ,0 0,24 x 2x
f xx 1 , x 2,
2 x 4
{ }: R 0,1g R
: ( )
+=
x 0
x g x 2xlim 5
3x ( ) ( ) ( )+ = +g x 3 g x f x x R .
: i. ( )
x 2limf x . ii. ( )
x 0limf x . iii. ( )
x 0limg x .
iv. ( )x 3
limg x .
28. f ( ) = + + 3xf x 3ln2x e 4x 2 . i. f. ii. : ( )
x 0limf x ( )
+xlim f x .
iii. ( ) =32f x e .
iv. : ( ) ( ) ( ) + + + =
23 12 2 63ln4 3ln 2 2 4 1 e e 8
-
- 9
45
29. : Rf R : ( ) 213 x 2xf x x
2, x R .
( ) ( )+ + = 24f x 3f x 1 2x 2013 , x R . i. ( )
x 0limf x .
ii. ( )f 1 . iii. f ( ) = g x x 1 ( )0x 0,1 .
30. : Rf R : ( ) = + + 2 2 2x x f x 1 x
x R
1A 0,2
.
i. . ii. =1 =1 f.
iii. : ( ) x 0
f xlim
x.
31. f ( ) + =3 x x
xx 2 3 2 4f x
2.
i. f . ii. ( )
xlim f x .
iii. ( )+xlim f x .
iv. ( ) = f x R R .
-
9
46
. 0f x( )
1. : i) 43x)x(f ++= x0 3= ii)
3 2x)x(f = 0x 0 =
iv) x3xx)x(f += x0 0= v) 11x)1x()x(f += x0 1=
vi) 3 4x)x(f = 0x 0 = vii) 3 x)x(f = x0 0=
2. x0
1. ( ) 0f x x 1 x , x 1= = 2. ( ) 0x x
f x ,x 02 x
= =+
3. ( ) 0xf x ,x 0
1 x= =
+
4. ( ) 0f x x x , x 0= = 5. ( )3 2
0f x x , x 0= = 6. ( )2
0x , x 0f x , x 0x
0 , x 0
= =
=
3. 0
) 12x , x 0f (x) x
0, x 0
==
, 0=0 )
13 xx e , x 0f (x) 0, x 0
1 12 xx e , x 0x
, 0=0
4. f 0 2 4f (x) x x x , x R ,
f 0.
5. f , : 22 xx)x(fxx + x R , f 0x 0=
f 0( )
6. f , g , f()=g() f(x)+x2 2g(x) , x R, + g ( ) f ( ) 2 =
7. f x0 ,
g 00 0 0 0
f (x) , x xg(x)
f (x ) (x x ) f (x ) , x x
=
+ > x0 .
-
9
47
8. g,f : 2g(x) f (x) g(x) (x 1) + Rx g '(1) 2= . f 1x 0 = )1('f .
9. , x = 0 ,
23x 1,x 0f (x)
ax ,x 0
+
-
9
48
18. 0 f ,
3 2 2 3
xf x 2f f x 3f f x 2f
x +
( ) ( ) ( ) ( ) ( ) ( )lim
19. > 1 2
0 0 1 2h 0
f x h f x h h h mh
+ + + + + =
( ) ( ) ...lim ,
.m R 0 1f (x ) m=
20. f(3) = 10 f (3) = 6 , 3x
lim2
2
(f(x)) -100x - 9
.
21. f 2)1('f)1(f == . :
)2
2x 1
f x 4x x 2
+ ( )lim )
x 1
xf x 2x 3 2
+ ( )lim
22. f Rx 0 , :
0
0 00 0 0x x
0
x f x x f x f x x f xx x
=
( ) ( )lim ( ) ( ).
23. f R R: *+ 1)(f)(f == :
x
f x f xx
( ) ( )lim
24. f R, x0 0= f(0)= 0 ,
: )0(fx1fxlim
x=
+.
25. f (). ( ) ( )x
xf f xlim
x
26. f (). ( ) ( )x
x f f xlim
x
2 2
27. f (). ( ) ( ) ( )x 0
f x f xf lim
2x+
='
28. : f 0x , :
) 0 0 0h 0f x 2h f x 2f x
h+ =( ) ( )lim '( ) ) 0 0 0h 0
f x h f x 3h 4f xh
+ = ( ) ( )lim '( )
)2 2
0 00 0h 0
f x h f x 2f x f xh
= ( ) ( )lim ( ) '( ) v)2
0 0h 0
f x h f x 0h
+ =( ) ( )lim
III. 0f x( )
-
9
49
29. f R i) )(f2)(f
x)(fx)x(flim 2
22
x=
ii) g R , :
)(g)(f)(f)(gx
)(f)x(g)x(f)(glimx
=
IV.
30. f x y R, :
22 y)x(f)yx(fy)x(f ++ . f x R0 .
31. f f(x + y) = f(x)f(y) , x, y f(0) 0 . f 0 ,
f (x) = f (0)f(x) , x .
32. f, g R : i) f x y f x f y+ = ( ) ( ) ( ) x y R, . ii) f x 1 x g x= + ( ) ( ) x R iii)
x 0g x 1
=lim ( )
f x R0 .
33. f : 2f 2 h 1 2h h 3h+ = + + +( ) , h R ,
: i) f 2 1=( ) ii) f 2 3 =( )
34. f 1 f( x ) = f( x)+ f ( ),
f xf x f 1 x= + ( )( ) ( )
35. A f xy f y f x x y 0 f 1 1 2= + =( ) ( ) ( ), , ( , ) , ( ) / f
(0, )+
36. f : R R , : yxf (x y) e f (y) e f (x)+ = + , x,y
f (0)=2
) f (0) 0= 2x ,f (3x) 3e f (x)= x R
) x 0 x 0
f x f 2xx 2x ( ) ( )lim ,lim ) f(x)
-
9
50
37. N
1. f x x x=( ) 2. x -1 x +1f(x)= +x +1 x -1
3. x
f(x)=x
4. f(x)= x
5. x -1 x +1f(x)= +x +1 x -1
6. 43f(x)= x
38. f(x) = 2
2.
x x x, x
-
9
51
4xy (x)+2y (x)- y(x)=0 ,x R
46. f(t)=e-t(t), , R* , :
f (t)+2 f (t)+(2+2)f(t). (
.)
47. : y axay (x)=e y (x)-3y (x)+2y(x)=0 , x R .
1 1 2 2g(x)=c y (x)+c y (x)
48. f R ( ) ( ) + =2 2 4f ( x ) f ( x ) x
f (0)= 0
49.
) xf(x)= x , x >0)
2x +3xf(x)=x
) xexf(x)= x ,x >0) ( ) ( )xf (x) x ,x 0,=
50. f : R R . g ,g ) 2g(x)= f(3x)+f (2x)
) ( )3 4 2g(x)= f x +x +1 ) 41+f (x) f(x)g(x)=e -x ,x >0
51.
) f(x)=(2x)(2x) ) f(x)=4(ln56xe ) ) f(x)=xx , x>0
) f(x)=1x(1+x) ) f(x)=
4 2
21 ln x+
) f(x)=3 x 2 , x [2,11)
3 x 2 , x [11, )
+ +
52. R f ( )6 2 3f(x )= f x f (x) 0, f(1)
53. / R : x
1 1 1- =f(x) g(x) e
: 2 2g(x)-g(x) f(x)- f(x)=
[g(x)] [f(x)]
54. = + = >5 3x t 5t , y 5t ,t 0 dxdy
55. ) f(0) = f (0) = 0 ) = h(x) f(x) f (x) 0, x R
-
9
52
= +
h (x) f (x) f (x), x Rh(x) f(x) f (x)
56. * 1x + 2(2x) + 3(3x) +.+(x) x x 1,2,3,. ( ),
: 1 + 22 + 33+.+ 1 .
57. A f x : f(ex-1)=(x) , f (1) , f (1)
58. f(x)=x(x-1)(x-2) (x-100) , f (0)
59. A f(x)=(x-)3(x-)5 , f (x) 3 5= + x x f(x) x- x-
60. f , g , g(x) 0,
x. h(x)= f (x)g(x)
. A 0h (x ) 0 = :
h(x0)= 00
f (x )g (x )
, g(x0) 0
61. f . : ) f , f ) f , f
62. f ,
g(x) = f(f(x))-f 21
x 1
.
63. f,g x0 f(x0) 0
G(x) = f(x)g(x)e
G(x0) = 0 g(x0) = f(x0) g(x0).
64. :
) f(x)=ex , f ()(x)=ex ) f(x)= (x) , f )(x)=(x+ 2
)
65. ) xf(x)= , x R2
) *1f(x)= , x Rx
.
66. (x) = [(x)]2 x .
-
9
53
o (x)- (x) = 2x2-7x+7, x .
67. P(x) : ) ( )xe P(x) =e x x(x-1) , x ) 1 [ P (x)]2=P(x) , P(1)=0
) 1 P(2x)= P (x) P (x)
68. f(x) = x2+x+, 0, 1,2, : ) f( 1)+ f( 2) = 0 ii) f( 1)f( 2) 0 ) 1/f( 1)+ 2/f( 2) = 1/.
69. : ) = 1 + 2x +3x2+.+x-1, x * ) = 2x + 4x3 + 6x5 + + 2x2-1, x *. 70. f(x) 2, : ) f(x) = (x-)2(x)
f() = f() = 0 ii) ,
(x-1)2 f(x) = x+2- x+1+2 2.
71. f(x)=x+ -1x-1 + +1x+ 0 , 0, 0 0 1, 2, , . :
i) i1 2
f (x) 1 1 1= + + ....... + , x i=1,2,3, .... ,f(x) x- x- x-
ii) f(x)f (x) < [f (x)]2 , x
iii) N 1 2
1 1 1+ + ....... +
iv) f(1)= f (1)=0 f (1)0 1 2 1 3 1
1 1 1+ + ....... + 0 - -
=
72. : ) f: (0,) R , f(x) = x, ) f -1 (-1,1
[f -1(x)]= 2
1 , x1 x
(-1,1).
73. f: ( ) ( )0, 0,+ + , f(x)f 1x
= x,
>0 :) f 1x
= 1/xf(x), x 0.
) [f(x)f 1x
]=0, x0.
IV.
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9
54
74. f(x) = , x ,2 2
1f
75. :f R R 1-1 R ( ) =f R R
R 1f (f ( )) 0 = 1f
0x =
76. f,g 1-1 R.
R = = 1 1f (x) g (x), g (x) f (x), x R
( ) ( ) + = + f g g f (x) x f g (x)
77. f(x)=5ex+x-2 . N
) f() 0 ) ( )1 1f (f ( )) f ( ) =
) ( )1 1f (3) 6
=
78. f ( )+0, , ( )0, + = + .f( ) f( ) f( ) ) =f(1) 0
) = > .x f (x) f(x) x f (1), x 0
79. f R , + = +xf(x ) e f( ) e f(x) + .+ , R
) f(0)= - = 0 ) xf (x)= f(x)+ f (0)e + x
80. f(x)=exln(2-ex) x0=0
81. Cf x0:
) f(x) = (x2-1) x , x0 = 4 ii) f(x) = x
x-xx, x0 = ) f(x) =
ln xx
, x0 = 1
iv) f(x) = xxe
, x0 = 0 v) f(x) = xx , x0 = 12
V.
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9
55
82. f(x) = x3-4x, Cf xx.
83. Cf f(x)=2x 5
x 1
,
M (x0,f(x0)) f(x0) + f(x0) = 4
84. f: f (x) + f (x) = 1 + x + x : x .
Cf (0, f(0)) (f(1), 2).
85. f(x)=x3+x2+x (0,1)
86. Cf f(x) = lnx
87. Cf f(x)=x3 o (0,-16).
88. Cf f(x)= xx 2+
,
(0,1).
89. f: R R : 3f(x+1) 2f(x-2) = x2 + 14x 5,
xR. ) f. )
Cf , (1,-14
), .
90. f : f (lnx) = x . lnx x, x > 0.
. Cf .
. Cf x0 = 0
91. (,) 2
c: = x2. ii) : = - 14
,
.
. 0
-
9
56
92. f(x)= 21 x ,
x x (,f()) , 0 Cf
) Cf (,f())
) :
i) =3 Cf
ii) =1 Cf
93. f (x) = x2 x + 1 ( ),
: ) = 3. ) 45 xx. ) y = x +3 .
) y = - 21 x + 3.
) xx. ) yy.
94. f (x)=x+ln(x-3)2 i) =3
ii) 45
iii) yy
95. y=3x-8 f(x)= x+ln(x-3)2
96. f 2f(x)=1+xf3(x) , x y=-x+2 Cf
97. f(x)=x2+x+2 : y=-x+ Cf
98. 0 1< , : y=x Cf f(x)=x.
.
. Cf
IV.
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9
57
99. g(x) = f(x)(x), 0, f R f(x) 0 xR. (x0,0) Cf Cg,
Cf Cg (x0,0).
100. g(x) = 1x 1
f(x) = x2 x +9. ,R,
Cf Cg 2.
101. , , f(x)=+xx g(x)= x2+x+2
(1,2)
102. f,g, : f R f(x) 0 , xR
i) R ii) g(x) = f(x)(x), xR
) [ (x)] 2 + [(x)] 2 = 1, xR (x0,0) Cf Cg, Cf Cg
103. N f(x)= 1x
g(x)=x2-x+1 ,
104. A f , f(x) 0 x g(x)=f(x)3x , f g
V.
105. g(x)=2x2x+ f(x)=23x 2x 4
x + .
, Cf Cg
2.
106. 4(1-x)e2x = 1 (0,1). ii) N Cf Cg , f(x) = ex g(x) = x .
107. = xf(x) e = g(x) lnx fc
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9
80
gc x x, f g
108. f(x)=ex g(x)=-x2-x. N Cf (0,1) Cg
109. Cf , Cg f(x)=x g(x)=x3-3x2+4x-1 , Cf , Cg
110. . f(x)=4-x2 g(x)=-x2 +8x-20.
V. 111. f : f (4) = 0 (f (x))2 = f (x) x R, ) f. )
Cf y = - x + 1
112. f [0, )+ x R = + +x 2 .f(e ) x 2x 3 fc A(1,f(1))
113. : = 2+1
f -1 , ( ).
x -x +1lim f +1 x -22- x
114. : =2-4 ( )g(x)= ln f(x) 0x = 2 f
2x f(x)-4 , x 2h(x)= (x -2)f (x)
6 , x = 2
115. f , g , h: . f(x) 0 , g(x)=f(x)h(x) x , f , g , h h2(x)=1-[h(x)]2,
Cf Cg .
116. f: x f(x)=f(+x) , . Cf
(-,f(-)) (+,f(+))
-
9
81
x=.
117. f f (1)=1 g g(x)=f(x2+x+1)-1 , x
Cf (1,f(1)) Cg (0,g(0))
118. f: : f (x) + f (x) = 1 + x + x x .
Cf (0, f(0)) (f(1), 2).
119. . cm r = 4-t2 , t o sec N
V t=1sec
120. x2 + y2 = 4. (-1, 3 ), 6 m/sec.
.
;
121. x >0 , (0,0), (4x,0) (0, x -2). x
2cm/sec, x = 9 cm.
122. 2m/sec 12m.
123. x2 + 2 = 2 xt
= .
t
0.
; 124. f(x) = x2 2x.
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9
82
T t0 (2,0) x 3 cm/sec.i) (t) t0. (t) x (t) 0 = , = f(x).
125. (t) = t40
1 19.2+,
t . (t) = N (t) = 10-3, .
126. 170cm
300cm. 7,2Km/h
127. 9 cm2. O
10 cm2/sec.
16 cm2.
128. 13m . 2m/sec. :
.
5m.
.T 12m.
129. f(x) =lnx, x0 (,ln), 0. Cf .
;
v = 2m/sec,
t t0
.
130. f(x) = x3, (,3), >0, yy v = 3 m/sec.
i) Cf . ii) Cf
.
-
9
83
iii) ( ) .
iv) t t0 yy
2m.
De L Hospital
131. ) 2x 0
x x ln(1 x)lim
+ +
) x 4
2limln( 2 )
)
x x
x 0
e e 2xlim1
v) x x
x 0
e elimx
132. :
) 4
2x 0xlim
x +2x-2 )
2
xx 1ln xlim
e ex ) 2x 0 x
x-xlimxe -1-x- 2
) 2x 0
xx-x+1limx
) 2
x 1x 1limln x
133. )
3
xxlim
e+ ) 1x 0
x
ln xlime
) xx
xlim , 0, 0e
+ > > v)
2
3 2x
x ln xlimx x x 1+ + + +
134. :
) x
ln( x 1)limln( x 2)+
++
) x
x
elimx
+ )
2
xx
x x xlime x 2++ + + +
135. )
x
1lim ln 1x+
+ ) ( )
1lim ln ln 1
x x x ) ( )x 0lim ln x+
136. :
. 00
.
. 0 ( )
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9
84
) 32x 0lim (x ln x)
+ ) 2
1x
2x 01lim ex
) 12x 0lim ( xln x)
+
137. i)
x 0
1lim
ii) ( )xxlim e ln x+ iii) 2 2x 0
1 1lim
iv) x 0
1 1lim2 1
138. :
) x 1
1 1limln x x 1
) 2 xxlim (x ln x e )+
+ ) 2 x
x 0 x elim -
x x
139.
) ( )x 0lim
+
)
x 0lim
+
)
11
x 1lim
v)
2
x
3lim3
+
+
v) ( )3
x 0lim 3 2
+
140. 2xf (x) ,0
g(x)0, 0
< < =
f R
f(0)= .f (0) f (0) 0 = = Na f g
141. f R f(0)=0.N xx 0
f (x) f ( x)lim 2f (0)e 1 =
142. f R f
=0 . f(0)= f (0) = 0 f (0) 0,
2
xx 0
f (x)lim 1(e 1)f (x) + =
f (0) 2 =
143. x ln x x 1,0 x 1f (x) 1 x2, x 1
+ < =
= f
. ( ) ( )
V. ( ),1 , 000
-
9
85
1f (1)2
=
144. 2x 2(1 e ) ln x , x 0f (x)
0, x 0
==
)
2x
2x 01 elim
x
2 2x 0lim(x ln x )
) f
=0. ) f (0,0)
145. f 0 =0 (0) 0.=f f (x)
x 0lim x 1
+=
146. f : R R f(x)>0
x
.lim (f (x) f (x)) l R+
+ = lim ( )x
f x l+
= lim ( ) 0+
=x
f x
147. f ( ) 0, f
2x1 1 f ( )lim
(x )f ( ) f (x) f (a) 2f ( )
=
148. x x xlim f (x) lim f (x) lim f (x)+ + +
= = = + x
f (x)lim 1xf (x)+ =
2
x
x f (x)lim 0xf (x)++ =
149. f: xf(x)+ex=f(x)x+ex x . f(0).
150. f: (1-x)f(x) = ln(1+x)-x x -1. f(0).
151. i) N 1+lnx = x (0,2). ii) Cf f(x) = xlnx+x+2, x(0,2) (2,5) f((0,2)).
152. f : . : 2h 0
f (x 2h) 3f (x) 2f (x h)lim 3f (x)h
+ + = .
ROLLE
-
9
86
. ROLLE
I.
153. Rolle ) f(x)=2x3+x2-8x+1 , x[-2,2]
) f(x)=2x+2x-1 , x[0,] ) f(x)=x2-4x+|x-2|+3 , x[1,3 ]
154. f(x) = 29x 14x 9, x 1
x+1 , x 1
+
.
.ROLLE [0,2] f.
155. f(x) = 2 x , x 0 x +x , x 0
. , R
f . ROLLE [-,1] x0(-,1) f( x0) = 0.
156. 2
3
x a x 0f (x)
x (-1), 0
+ =
+ > , R
f Rolle [-1,1] 157. : i) f(x) = xx
. ROLLE [- ,2 2 ] ii) xx = 1
(- ,2 2 ).
158. f : [,] (,) f() = f() = 0 :
a) g(x) = f (x)x c
c[,] . ROLLE
[,]. b) c[,], c0(,) , Cf (c0, f(c0)) (c,0).
159. f: [a,]R, [,] (,). : I )
)x)(ax(e)x(g )x(f = Rolle [,]
) (,) , f ()= 1 1a -
+
-
9
87
160. f,g : [,] f() = f() = 0 f(x) 0 x(,)
: a) h(x) = f(x)e-g(x), x[,] . ROLLE [,]. b) x0(,) , Cg (x0,g(x0)) : f(x0).x - f(x0). + = 0.
161. f(x) = x (x-1) ,*.
x0(0,1) , f(x0) = 0 (x0,0) A B =
(0,0) (1,0).
162. f [,]
f() = f() f() = f() = 0, x1, x2(,) , f (x1) = f (x2).
163. f [,], (,) 0[,],
x0(,) , : 02 20
f (x )f ( ) f ( )2x
=
.
164. f ,g [,], (,) f() = f()eg()-g(), x0(,) , : f(x0) + f(x0).g(x0) = 0.
165. f R 10f(x)6f(5)+4f(3) x, f() = 0. 166. f [2,3] f(3) f(2) = ln3-ln2,
(2,3), f() = 1
.
167. .
168. f [,], (,). x0(,) , : f(x0) = 0
0
f (x ) f ( ) .x
169. f [0,2] (0,2),
(0,2), f() = f(2-). 170. f [,], (,)
f() - f() = 2-2. x0(,) , : f(x0) = 2 x0. 171. f: [1,4] R f(1) = 2
-
9
88
f(4) = 8. Cf
172. f: [,] R [,] , (,)
f(x) 0 , f () 1 1(,)f () - - = +
173. ( ) ) x1 , x2
f
f
) 1 , 2
f
f
) f x1 , x2 , , x
f -1
: ( ) f :
F
F (x) f (x) = , x
. ROLLE
f F f F
0 c f(x) f(x)
1 x f(x) f(x) 12
f2(x)
x +1x
+1 f(x) f(x)
1+1
f+1(x)
1x
2 x f (x)f (x)
2 f (x)
-
9
89
x -x f(x) f(x) -f(x)
x x f(x) f(x) f(x)
ex ex ef(x) f(x) ef(x)
1x
ln|x| f (x)f (x)
ln|f(x)|
x x
ln f(x) f(x)
f(x)ln
f F f F
f(x)g(x) + f(x)g(x) f(x)g(x) f(x) +x f(x) x f(x)
2
f (x)g(x) f (x)g (x)g (x)
f (x)
g(x)
2
f (x) xf (x)x
f (x)x
[f(x)]2+f(x) f(x) f(x) f(x) [f(x)+ f(x)g(x)]eg(x) f(x)eg(x)
174. 0 , , R 03 2 + + = . 2++=0
( 0 , 1) .
175. + ( -1) =0 ( 0 ,1 )
176. 1 02 1..... 0, 1 3 2 1
+ + + + + = +
*,
x+ -1x-1 +..+2x2 + 1x + 0 = 0, (0,1).
177. 8x3+9x2-6x-2=0 (0,1)
178. (x2-4)x+2xx=0 (-2,2)
179. f [,] , >0 (,)
f() f()=
) xf(x)=f(x)
(,) )
Cf
180. ex =1 , ex = -1
-
9
90
181. 24x+xe = 2x 2 + 4 =
182. 3 3 + =0 (-1 , 1 ) 183. 2 + + =0 184. f : R R R
f (x) 0 R .
f (x) =0 .
185. 0, x4+x3+2x2 +x+ = 0 .
186. 4x3 + 18x + = 21x2 (1,2).
187. f(x) = 3 22 7x - x +3x+3 2
, R.
. 188. x3+x2 +x+ = ex
. 189. x4+2x3+3x2 x + = 0
, R.
190. x2-2x+2ln(x+1)=0
191. x4-x3+5x2+x+=0 ,
-
9
91
) : (x+3)1996 = (x+1)1996 +16499.
193. 4 + 3 +32 + + =0
, 2 > 8 194. (x) = (2x-1)(2x+1)(x2-3x).
(x) = 0,
..
195. f(x) =
3
2x+2, x -1x - x, x >-1
. f
... [-3,2] , (-3,2) ... 196. f f(x) = x(1-lnx). f
... [1,e] , (1,e) ... 197. f [,], (,)
f(x) 0 x[,]. ... [,] f(x) = lnf(x), (,)
: f( )( )f ( )f ( ) f ( )e
=
. ..
198. f [1,3] f(1) = 2005
1 1f (x)2 2
x(1,3), 2004 f(3)2006.
199. f(x) R f(0) = 0, : f(1) f(1) f(0).
200. f [1,5] f(1) = -2 f (x) 2 x(1,5), : -10 f(5) 6.
. ..
-
9
92
201. ) f [,] : f()- f()f ()<