www.mathjazz.com

www.mathjazz.com 68

x2 x1

f (x1)

f (x2)

x

y

4

1. ,

. 2. . 3.

. 4.

5. . 6.

.

7. .

.

f :

, : 1 2x ,x 1 2x x< 1 2f (x ) f (x )< 1 2x ,x 1 2x x> 1 2f (x ) f (x )> .

f ,

f

, : 1 2x ,x 1 2x x< 1 2f (x ) f (x )> 1 2x ,x 1 2x x> 1 2f (x ) f (x )< .

f

, f .

, : 1 2x ,x 1 2x x< 1 2f (x ) f (x ) 1 2x ,x 1 2x x> 1 2f (x ) f (x ) .

x2 x1 x

y

f (x2)

f (x1)

Tom RaikHighlight ,

Tom RaikHighlight ,

www.mathjazz.com

www.mathjazz.com 69

f , f . , :

1 2x ,x 1 2x x< 1 2f (x ) f (x ) 1 2x ,x 1 2x x> 1 2f (x ) f (x ) .

f , f . f

. f f , , , f . f . f f , , , f . , 2f (x) x= : [0, )+ , 1 20 x x <

: 2 21 2x x< 1 2f (x ) f (x )< . ( ,0] , 1 2x x 0<

:

2 10 x x < 2 22 10 x x < 1 2f (x ) f(x )> .

f , x A f (x) c= , c .

. ; . f ,

, : 1 ()

1 2x ,x A 1 2x x< 1f (x ) 2f (x ) :

1 2f (x ) f (x )< , f , 1 2f (x ) f (x ) , f , 1 2f (x ) f (x )> , f , 1 2f (x ) f (x ) , f , 1 2f (x ) f (x )= , f .

O x

y=x2

y

Tom RaikHighlight

Tom RaikHighlight . , '

www.mathjazz.com

www.mathjazz.com 70

48 f 3f (x) x x 2= + + f fA = . 1 2 fx ,x A = 1 2x x< (1). (1) ( ) 3 31 2x x< (2). (1) (2) :

3 31 1 2 2x x x x+ < +

3 31 1 2 2x x 2 x x 2+ + < + +

1 2f (x ) f (x )< , f fA = . 49 f xf (x) e= f fA [0, + )= . 1 2 fx ,x A [0, + ) = 1 2x x< (1). (1) 1 2 1 2x x x x< > (2). xy e= (2) 1 2x xe e > 1 2f (x ) f (x )> , f fA [0, + )= . 50 f xf (x) e 2x 1= + . i. f. ii. f (0) . iii. fx A fC : . x x , . x x .

f fA = . i. 1 2 fx ,x A = 1 2x x< (1). (1) :

1 2x xe e< (2) xy e= 1 2 1 22x 2x 2x 1 2x 1< < (3).

(2) (3) : 1 2x x

1 1e 2x 1 e 2x 1+ < + 1 2f (x ) f (x )< ,

f fA = . ii. 0f (0) e 2 0 1 1 1 0= + = = . iii. f f (0) 0= : . x 0 f (x) f (0) f (x) 0< < < . . x 0 f (x) f (0) f (x) 0> > > .

www.mathjazz.com

www.mathjazz.com 71

: x ( , 0) fC x x . x (0, + ) fC x x . 2 ()

1 2x ,x A 1 2f (x ) f (x )< f :

1 2x x< , f , 1 2x x> , f

1 2x ,x A 1 2f (x ) f (x ) f :

1 2x x< , f , 1 2x x> , f .

51

f f(x)= 1 xln1 x+ . .

: !!! 1 2x ,x A . 1 2f (x ) f (x )< 1 2x x<

1 2x x> , . f x : 1 x 01 x >+ (1-x)(1+x)> 0 1- x

2> 0 x2

www.mathjazz.com

www.mathjazz.com 72

!

0 = f . ( ) f , :

1 2

1 2

f (x ) f (x )x x

= , 1 2x x , ,

( )1 2 1 21 2

1 2

x x g(x , x )g(x , x )

x x = = , 1 2g(x ,x )

52 f f (x) x 2= + . f x : x 2 0 x 2+ , f =[-2, +). 1 2 fx ,x A 1 2x x . ( ) f , :

1 2

1 2

f (x ) f (x )x x

= =1 2

1 2

x 2 x 2x x+ += =

( ) ( )( ) ( )

1 2 1 2

1 2 1 2

x 2 x 2 x 2 x 2

x x x 2 x 2

+ + + + + = + + +

( ) ( )( ) ( )

2 2

1 2

1 2 1 2

x 2 x 2

x x x 2 x 2

+ += = + + +( ) ( )

( ) ( )1 21 2 1 2x 2 x 2

x x x 2 x 2

+ + = + + +

( ) ( )1 21 2 1 2x x

x x x 2 x 2= = + + + 1 2

1 0x 2 x 2

>+ + + , f

f =[-2, +). . ; 1 f (g(x)) f (h(x))< , x , : f x . f g(x) h(x)< , x ... f g(x) h(x)> , x ... f (g(x)) f (h(x)) f (g(x)) f (h(x))> f (g(x)) f (h(x)) . 53 f xf (x) e 2x= + . 2 2f (2x 3x 3) f (x x) + < + .

Tom RaikHighlight .

www.mathjazz.com

www.mathjazz.com 73

f . f fA = . 1 2 fx ,x A = 1 2x x< (1). (1) : 1 2x xe e< (2) xy e= . (1) (2) :

1 2x x1 1e x 1 e x+ < + 1 2f (x ) f (x )< ,

f fA = . f fA = ,

2 2 2 2f (2x 3x 3) f (x x) 2x 3x 3 x x + < + + < + 2x 4x 3 0 (x 1) (x 3) 0 1 x 3 + < < < < , x (1, 3) .

2 g(x) h(x)= , g hx A A , , : i. g(x) h(x)= , g hx A A . ii. g(x) h(x)= !

f (x) g(x) h(x)= , g hx A A , . iii. , ,

! g(x) h(x)< g(x) h(x) g(x) h(x)> g(x) h(x) ,

g hx A A , , . 54 5 x ln x 2 = + . , , ! . ! f (x) 5 x ln x 2= . f fA (0, 5]= . 1 2 fx ,x A (0, 5] = 1 2x x< (1) 1 2x x > 1 2 1 25 x 5 x 5 x 5 x > > .(2) (1) y ln x=

1 2 1 2ln x ln x ln x ln x< > 1 2ln x 2 ln x 2 > .(3) (2) (3) :

1 1 2 2 1 25 x ln x 2 5 x ln x 2 f (x ) f (x ) > > , f . f (1) 5 1 ln1 2 0= = . : x 1 f (x) f (1) f (x) 0> < < .

www.mathjazz.com

www.mathjazz.com 74

C f

f (x0)

f (x)

Ox

y

x0 x

y=x2+1 1

O x

y

y=|x1|

1 O x

y

x 1 f (x) f (1) f (x) 0< > > . x 1 f (x) 0 f (1) 0= , 5 x ln x 2 = x 1= .

f :

0x A () , 0f (x ) , 0f (x) f (x ) x A .

0x A () , 0f (x ) , 0f (x) f (x ) x A .

2f (x) x 1= + 0x 0= , f (0) 1= , f (x) f (0) x .

f (x) | x 1 |= 0x 1= , f (1) 0= , f (x) f (1) x .

C f

f (x0) f (x)

O x

y

x0 x

Tom RaikHighlight

www.mathjazz.com

www.mathjazz.com 75

O

y=x

2 5/2 3/2 /2

/2

1

1

y

x

O x

y

y=x3

f (x) x= , y 1= , 2k

2+ , k ,

y 1= , 2k2

, k Z , 1 x 1 x R .

3f (x) x= , , .

, , , . () () f () f.

; f : 1 fC f, : , , ( )0 0x , f(x ) , , min 0y f (x )= .

0 minf (x ) y= ()

, , ( )0 0x , f(x ) , , max 0y f (x )= .

0 maxf (x ) y= ()

www.mathjazz.com

www.mathjazz.com 76

!

!

: max 1y f (x )= , min 2y f (x )= .

1 maxf(x ) y=

1x

min 2y f (x )=

2x

2 , , . , : 1 2y , y 1 miny y= 2 maxy y= , )1 2y , y )1y , + 1 miny y= , ( 1 2y , y ( 2, y 2 maxy y= , ( )1 2y , y ( ), + ( )1y , + ( )2, y .

f : 1 . 1. 2 . .

:

f [ ], , minf () y= maxf () y= , f [ ), , minf () y= maxy , f ( ), , ( ), + , ( ), + ( ),

miny maxy ,

f ( ], , miny maxf () y= , f [ ], , minf () y= maxf () y= , f [ ), , miny maxf () y= , f ( ], , minf () y= maxy . f ( ), , ( ), + , ( ), + ( ),

miny maxy .

Tom RaikHighlight

www.mathjazz.com

www.mathjazz.com 77

2f (x) x x= + + , 0 x , : 0> , : :

, 2

, +2

,

, +4

, 2 -4 = ,

0 x 2= miny f

2 4 = = ,

2 -4 = .

www.mathjazz.com

www.mathjazz.com 78

x 2 0 1 y 0 . 2x 2 (1 y) = y 1 2x 2 (1 y)= + y 2 .

3. fx D x 2 2 22 (1 y) 2 (1 y) 0+ , y , ff (D ) ( , 1]= .

4. : f , maxy 1= . 56 f f (x) x 1= 4 x 3 . 1 2x ,x [ 4, 3] 1 2x x< 1 2x 1 x 1 < 1 2f (x ) f (x )< , f A [ 4, 3]= , [ ] [ ]f (A) f ( 4), f(3) 5, 2= = . miny 5= maxy 2= . 57 f 2f (x) x 4x 3= + , x , .

: 02 2

=1>0 4x 2

2 2 1 4=( 4) 4 1 3 4

= = = = =, :

: ( ], 2 [ )2, + ,

[ )1, + 0x 2= min 4y 14 1= = . 58

2x 1 10 x 0

f (x)x 1 0 x 10

+ = + .

. 21f (x) x 1= + , [ ]1x A 10, 0 = 2f (x) x 1= + , [ ]2x A 0, 10 = . x 0 x y . 1 2f (0) f (0)= . 1f (0) 1= 2f (0) 1= ,

2x 1 10 x 0f (x)

x 1 0 x 10

+ = + .

2y x 1= + :

www.mathjazz.com

www.mathjazz.com 79

02

=1>0 x 0

2 4= 4 1 1 4

= = = = ,

( ], 0 [ )0, + ,

[ ]1A 10, 0= , [ ] [ ]1 1f (A ) f (0), f(10) 1, 101= = .

y x 1= + [ )1, + [ )1 2x , x 1, + 1 2x x<

1 2x 1 x 1+ < + 1 2x 1 x 1+ < + 1 2f (x ) f (x )< ,

[ ]2A 0, 10= , [ ]2 2f (A ) f (0), f(10) 1, 11 = = . :

2x 1 10 x 0

f (x)x 1 0 x 10

+ = + :

[ ]1A 10, 0= [ ]2A 0, 10= ,

[ ] [ ]f (A) 1, 101 1, 11 1, 101 = = , miny 1= maxy 101= .

59

f 2 xf (x)2 x+ = .

24 22 f1f (D ) ,33

= , min1y3

= maxy 3= .

60

f 2

2x 1f (x)

x 5x 6+= + .

23 21 ff (D ) ( , 14 2] [ 14 2, )= + + , .

www.mathjazz.com

www.mathjazz.com 80

y

5

2

1

-3 -2 -1 5 x

61 f x 1f (x) e = , [ ]x 2, 101 f [ ]fA 2, 101= . [ ]1 2 fx , x A 2, 101 = 1 2x x< (1). (1) 1 2 1 2x 1 x x 1 x 1 < > (2). xy e= (2) 1 2x 1 x 1e e >

1 2f (x ) f (x )< , f [ ]fA 2, 101= , 2 1

miny f (2) e e= = = 101 1 10maxy e e= = .

127 i. f (x) 1 x= ii. f (x) 2ln(x 2) 1= iii. 1 xf (x) 3e 1= + iv. 2f (x) (x 1) 1= , x 1 . 128 f R . f [, ] , > 0, f [- , - ]. 129 Cf f . : i. f, ii. f, iii. f, iv. f, v. f, :

[- 1, 0) yx

= [0, 2) y = x2.

130

i. > 0, 1 2

+ .

ii. 1f (x) xx

= + x > 0.

www.mathjazz.com

www.mathjazz.com 81

131 f, g ,

x . 1 1f g+

. 132 f, g R, ( ). i. fog . ii. fof gog. iii. ( )f (x) ln ln x= , x > 1. 133 f f (x) 1 x= , x[0, 2]. 134 f(x)=x(x-2), x [0, 2]. i. f (x) 0 x Df=[0, 2]. ii. f(x)=(x-1)2 -1 f

[ 1,0] . iii. f.

v. x y=0 3y4

= . 135

i. eln xx

= , x 0> .

ii. eln xx

> , x 0> . 136 i. ln x 1 x= . ii. ln x 1 x> . 137 i. xln x 2e x 1 2e+ + = . ii. xln x 2e x 1 2e+ + < . 138

i. 10 x ln x 3 = . ii. 10 x ln x 3 < . 139 : i. x ln x 1+ = , ii. 3x ln x 3+ = .

www.mathjazz.com

www.mathjazz.com 82

140 : i. x2 x 11+ = , ii. x x x6 8 10+ = . 141

0 < < , : e e ln < . 142 < , : e e < . 143

0 < < , : ln > . 144

0 < < , : e x ,

2f (x) 3g(x)3f (x)

= . 149

x 5 3f (x) e x x x 1= + + + 3g(x) 2 x x ln x= . i. . ii. f (x) 0> g(x) 0> . 150 f 0 f (x) 1< < x , g 2

f (x)g(x)1 f (x)

= + . 151

i. x x x

x x5 3 4

x

3 4ln e e5

++ < .

ii. 2e ln(x e)x

=

www.mathjazz.com

www.mathjazz.com 83

152 f x (f f f f f )(x) x=D D D D . f (x) x= x . 153 f xf (x) e x 1= + . i. f .

ii. f (0) 0= . iii. f

x x . iv. x 2x 2x 1 ,

2f (x ) f (2x 1) 0 x. 154 f . f ( ) 0 = , f ( 3) f ( 4) 0 + < . 155 f ,g,h . f h x f (x) g(x) h(x)< < , : (f f )(x) (g g)(x) (h h)(x)< f (x) f (x y)< + , f . 157 f xf (x) ( 1)x 2 1= + + 0 1< . i. f . ii. f (x) 0= . 158 f ,g . f f (x) g(x)< x , (f f )(x) (g g)(x)

www.mathjazz.com

www.mathjazz.com 84

162 f f (x) 0> x , g f (x)g(x)

1 f (x)= + .

163 f , g : i. f (x) 3x 1= + , x [ 2, 2] . ii. 2g(x) 3x 12x 7= + . . 164 f , g : i. 2f (x) x 3x 2= + . ii. 2g(x) x 4x 8= + . . 165 f , g :

i. ( )2f (x) ln 1 x 1= + + . ii. x 4g(x)

x 3+= , x [ 2, 2] [4, 8] .

. 166 f , g : i. 2f (x) x 4 x 2= + . ii. g(x) x 3 x 2= + , x [ , 4] . . 167 f , g :

i. x 1 1 x 2

f (x)2x 1 3 x 4+ = .

ii. 2

2

x 1 1 x 2g(x)

x x 1 2 x 4 + = + <

.

. 168 f , g :

www.mathjazz.com

www.mathjazz.com 85

i. 2

2

x x 1f (x)x x 1

+= + + .

ii. 2

2

x 1g(x)x 2x 1

+= + + . . 169

f f (x) 1 x 4= + + . miny 2= . 170

f 2f (x) x 1= + . maxy 4= . 171

f 2

2

x x 1f (x)x x 1

+ += + + . , miny 2= maxy 2= . 172

f 24xf (x)x

+ = + , 0 > . , miny 1= maxy 4= . 173

f 2

2

x xf (x)x 1+ + = + .

, min maxy y 0+ = .

www.mathjazz.com

www.mathjazz.com 86

1. 2.

3.

4.

5.

6.

7.

8.

www.mathjazz.com

www.mathjazz.com 87

9.