of 38
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e 2007/2008
1
( )y f x=
,
,
. :
(1) () (2) (3) ()(4) (5) (6) ; ;
:
(7)
,
.
, , :
. .
( )f x
( )x
( )x
; ; .
:
: ( )3
23
xf x =
.
Milo
Stani
Digitally signed by Milo
Stani
DN: cn=Milo Stani,
o=Tehnika kola Uice,
ou,
email=milos.chane@gma
.com, c=YU
Date: 2010.02.07 00:16:25
+01'00'
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e 2007/2008
2
( 1 ) ()
x ( )y f x= ..
x
.
: ( >> )
( )3
23
xf x
x=
x
23 x , :2
2
3 0
3
3
x
x
x
: ( ) ( ) ( ), 3 3, 3 3,fD = + .
:
( ) ( ) ( ), 3 3, 3 3,x + .
?
:
( >> )
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e 2007/2008
3
( 2 ) :
'''' .
; ; ; ;
:
7/29/2019 3967 Tok Funkcije
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e 2007/2008
4
, , .
:
( )limx a f x=
x a=
.
:
'''' a
,
.
:
( )limx
f x b
= y b= .
7/29/2019 3967 Tok Funkcije
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e 2007/2008
5
:
:
( )limx
xk
= ( )lim
xn f x k x
=
y kx n= + .
:
(
), ,
0k=
: 0y x n= + .
: ( )3
23
xf x
x=
.
''''
: ( )3 3 ,
:
( )( )
( )
( )
( )
3
3
223 3
3 3 3 3 3lim lim
3 3 3 03 3x x
xf x
x
= = = = =
: ( )3x =
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e 2007/2008
6
( )( )
( )
3
3
223 3
3 3 3 3 3lim lim
3 3 3 03 3x x
xf x
x = = = = =
: 3x = .
:
( >> ) ,
( )+ ( ) ( )3 3 '' ''. (
: > )
''''
( )
3
3 3
22
3 3
3
1lim lim lim lim
33 3x x x x
x
x xf xxx
x x x
= = = =
0
1
x
0
1
0= =
:
( ( )limx
f x
= ).
''''
( )
( )
3
3 32
32
3
3
3
3 3 2
3lim lim lim lim33
1lim lim
3 3
x x x x
x x
xf x x xxk
x x x xx x
x
x
x xx x x
= = = = = =
= =
( )01
1 1
0 11
k= = =
( ) ( )
( )
3 3
2 2
3 2 3
2
lim lim 1 lim3 3
3lim lim
3
x x x
x x
x xn f x k x x x
x x
x x x x
x
= = = + =
+ = =
33x x+ 2 2
2
2
2 2
3lim
3 3
33
lim lim3
x
x x
x
x x
x
xxx
x x
= = =
= =
0
2
3
x
0
00 0
0 1
1
n= = =
: ( )1 0y x= + y x= .
( >>)
(
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e 2007/2008
7
( 3 ) :
:
= f(x) je ( -) :
( ) ( )f x f x =
:
= f(x) je ( )
: ( ) ( )f x f x =
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e 2007/2008
8
: ( )3
23
xf x =
( )( )
( )( )
( ) ( )
3 3
2 233
x xf x f x
xx
f x f x
= = =
=
: .
( >>)
( > )
( 4 ) :
x : ( ) 0f x = ; ( ) 0f x > ;
( ) 0f x < . :
( ) 0f x (-)=
:
( )
( )
0
0
f x()
f x
>
<
( )3
23f x
x=
''''
( )3
2
3
0 03
0
0
xf x
x
x
x
= =
=
=
, ( ) 0f x = 0x =
:
'''' -
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e 2007/2008
9
''''
: ( ) 0f x > ( ) 0f x < .
.
( >> )
:
( )( )( )
3 3
23 3 3
x xf x
x x= =
+
:
( )3 0 3 +
3x + +
3 + + +
3 + + + +
( )f x + +
: ( ) 0f x > ( ) ( ), 3 0, 3x
( ) 0f x < ( ) ( )3,0 3,x +
(
0x = ,
)
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10
( > )
( >> )
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11
( 5 ) :
:
( ),a b ( )y f x= .
( )1 2, ,x a b 1 2x< ( ) ( )1 2f x f x<
( )y f x= .
1 2
x x< ( ) ( )1 2f x f x>
( )y f x= .
'' 0x '', ( )0' x ( )
( )( )0 0,T x f x ,
, ( )0'f x k= , .
,
,
.
y kx n= + , ,
( )0'y f x x n= + , :
( )0' 0f x > ( )y f x= 0
( ): f x y
( )0' 0f x < ( )y f x= 0x
( ): f x y
( )0' 0f x = ( )f x= 0x
. , ( )
( )
7/29/2019 3967 Tok Funkcije
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12
, ( )y f x=
0 ( )0' 0f x = .
:
( )f x= 0
( )0' 0f x = :
( )0'' 0f x < () ( )y f x= 0
.
( )0'' 0f x > () ( )f x= 0
.
( )0'' 0f x = () ( )y f x= 0
. ( 0 )
:
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13
( )3
23f x
x=
.
,
( )'f x , , ''''
( )'f x .
( )( ) ( ) ( )
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )( )
( )
3 2 3 2 2 2 33
2 22 2 2
2 2 2 2 2 2 2 2 2
2 2 2 22 2 2 2
3 3 3 3 2'
3 3 3
3 3 2 9 3 2 9 3 3
3 3 3 3
x x x x x x x xxf x
x x x
x x x x x x x x x x x
x x x x
= = = =
+ + + = = = =
( ) ( )( )
( )
2
22
3 3'
3
x xf x
x
+=
'''' : ( )( )( )
( )
2
22
3 3'
3
x xf x
x
+=
( )( ) ( )
( )
( )( )
( )
2
22
2
2
1 2 3
3 3' 0 0
3
3 3 0
0 3 0 3 0
0 3 3
x x xf x
x
x x x
x x x
x x x
+= =
+ =
= = + =
= = =
'''': ( )1 2 30 ; 3 ; 3x x x= = = .
( )"f x .
( )"f x :
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14
( ) ( )( )( )( )
( )
( )
( ) ( )
( ) ( ) ( ) ( )( )( )( )( ) ( ) ( ) ( )
( )
2 22 2 4
2 2 22 2 2
2 22 4 2 2 4 2
42
23 2 2 4 2 2
42
93 3 9" '
3 3 3
9 3 9 3
3
18 4 3 9 2 3 3
3
x xx x x x xf x f x
x x x
x x x x x xx
x x x x x x x
x
+ = = = = =
= =
= =
( )( ) ( ) ( ) ( )
( )
( )
23 2 2 4 2
42
2
18 4 3 9 2 3 2
3
2 3
x x x x x x x
x
x x
= =
=
( )( ) ( )
( )
2 2 2 4
42
9 2 3 2 9
3
x x x x
x
+
2 2 42 27 9 6 2x x x x
=
+=
2 418 2x x+ ( )( )
( )
( )
( )
( ) ( )
( )( )
( ) ( )
2
3 32 2
2
3 3
2
3 3
2 27 3
3 3
6 9
3 3
6 9"
3 3
x x
x x
x x
x x
x xf x
x x
+= =
+=
+
+=
+
(
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15
( ) ( )( )
( ) ( )
2
2 3 32
6 3 9 3 18 18 63
63 3f x f
+ = = = =
3 6 3
6
6 62
3
2=
: ( )3
3 0
2
= <
2
3x = .
:
( )3
max 2
3 27 9 13 4
3 3 6 2 2y f= = = = =
, max1
42
y
=
2 3x = , 11
3; 42
E
.
( ) ( )( ) ( )( )
( )( ) ( )
2
3 3 32
6 3 9 3 18 18 33
263 3
f x f +
= = = =
: ( ) ( )33
3 02
f x f = = > ( )3 3x =
.
:
( )( )
( )
3
min 2
3 27 9 13 4
6 2 23 3y f
= = = = =
,min
14
2y = ( )3 3x = , 2
13;4
2E
.
, , ''''
( )( ) ( )
( )
2
22
3 3'
3
x xf x
x
+=
.
:
7/29/2019 3967 Tok Funkcije
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e 2007/2008
16
3 3 0 3 3 +
2x + + + + + +
3 + + + + +
3 x+ + + + + +
( )2
23 + + + + + +
( )f x + + + +
( )f x
:
( )f x ( ) ( ) ( )3; 3 3; 3 3;3x
( )f x ( ) ( ); 3 3;x +
( > )
7/29/2019 3967 Tok Funkcije
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17
( 6 ) ; ; :
( ),a b
.
( ),a b
.
:
( )y f x= ( ),a b
( ),x a b , :
( ) 0f x > ( ),a b ( )y f x=
.
( ) 0f x < ( ),a b ( )y f x=
.
( )0 0f x = ( )0 ,x a b ( )y f x= 0x
.
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e 2007/2008
18
; :
( )3
23f x
x=
''''
(
==
, ( )3
23
xf x
x=
0x = .
( )0f P.
( )3
3
0 00 0
3 0 3f = = =
( )0,0P .
'''' :( )
( ) ( )
2
3 3
6 9
3 3
x xy
x x
+ =
+
.
7/29/2019 3967 Tok Funkcije
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e 2007/2008
19
:
3 0 3 +
6 + +
29 x+ + + + +
3 + + +
3 + + + +
( )x + +
( )f x
:
( ) ( )3,0 3,x +
( ) ( ), 3 0, 3x
( )0,0P .
>> :
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e 2007/2008
20
( 7 ) :
( )
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e 2007/2008
21
:
:
() :
(
( )... 0 )
1:
2
3 2
1
2 3
xy
x x
+=
+
:
( )
( ) ( )( )
( ) ( )
( ) ( )
( )
3 2
2
2
2 3 0
2 3 0
3 3 0
1 3 1 0
1 3 0
0 1 0 3 0
0 1 3
x x x
:
x x x
x x x x
x x x x
x x x
x x x
x x x
+
+
+
+
+
+
( 2 >> )
: { }| 0 1 3fD x R x x x=
: ( ) ( ) ( ) ( ), 3 3,0 0,1 1,fD = + :
:
( )y f x= ( ) ( ) ( ) ( ), 3 3,0 0,1 1,x +
?
:
7/29/2019 3967 Tok Funkcije
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22
() :
( ( )... 0 )
2:
:3 2
2
2 3
1
x x xy
x
+ =
+
:
. ,
, ,
.3 2 22 3 0 1 0x x x x+ +
2 1 0x + R .
3 22 3 0x x x+
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23
( ) 3 22 3I x x x x= + :
( ) ( )( )3 22 3 1 3I x x x x x x x= + = +
( ( )I x 1 >>)
():
(: (-3); 0; 1;,
'' - '' ( x+3); x; (x+1); )
() ( )+
() , ( )
.
( )I x ; ()
( )+ ( ) ''''
''''
()
3 22 3 0x x x+
( )+
( '' ... 0 '') .. :
[ ] )3;0 1;x +
2
1 0x + R ,
3 22 3 0x x x+ [ ] )3;0 1; +
3 2
2
2 3
1
x xy
x
+ =
+: [ ] )( ) [ ] )3;0 1; 3;0 1;fD R= + = +
( -3) 0 1 +
+ +
1x +
3+ + + +
( )I x + +
7/29/2019 3967 Tok Funkcije
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e 2007/2008
24
:
3 2
2
2 3
1
x xy
x
+ =
+ [ ] )3;0 1;x + .
.
() :
( ( )... 0> )
3:
:2
2
4 19 12ln
4
x xy
x
+ =
:
:
2
2
4 19 12 04
x x
x + >
:24 19 12x x + 2 4x :
7/29/2019 3967 Tok Funkcije
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25
24 19 12x x +
: ( )( )2 1 2Ax Bx C A x x x x+ + =
( ) ( )
( ) ( ) ( )
( )
2
2
1,2
2
1,2
1,2
1,2
1,2
1 2
1 2
1
4 19 12
4 ; 19; 12 ;
4
2
19 19 4 4 12
2 4
19 361 192
8
19 169
8
19 13
8
19 13 19 13;
8 8
32 6;
8 8
4 ;
x x
A B C
B B ACx
A
x
x
x
x
x x
x x
x
+
= = =
=
=
=
=
=
+= =
= =
=
( )
2
2
3
4
34 19 12 4 4
4
x
x x x x
=
+ =
2 4x
:
( ) ( )2 4 2 2x x x = +
, :
( )
( )( )
34 4
40
2 2
x x
x x
> +
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26
:
- -23
42 4 +
( )4
4x +
3
4x + + +
2x + +
2x + + + + +
( )I x + +
: ( )3
2, 2,44
x
: ( )3
2, 2, 44
fD
=
:
2
2
4 19 12ln
4
x xy
x
+ =
( )
32, 2,4
4x
.
:
( )
7/29/2019 3967 Tok Funkcije
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e 2007/2008
27
:
( )limx a
f x
, ( )f x
a x a .
a , ( )f x
( ),a a + 0 > .
( )f x ( )limx a
f x
( )f x ( )limx a
x +
.
:
( ) ( )0
lim lim ...x a
f x f a
= =
:
( ) ( )0
lim lim ...x a
f x f a
+
= + =
0 > .
:
4:
( )3
23f x =
:
) ( )3
) 3
7/29/2019 3967 Tok Funkcije
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e 2007/2008
28
:
) :
( )
( )
( )
( )( )( )
( )
( )( )
3 3
3
2 22 0 03
2 3 2 3
220 0
2 32 3
220 0
3
2
3 3lim lim lim
3 3 3 3 3
3 3 9 3 3 3 3 9 3 3lim lim
3 3 2 33 3 2 3
3 3 9 3 3 3 3 9 3 3lim lim
2 32 3
3 3 9 0 3 3 0 0 3
2 3 0 0
x
x
x
+ = = =
+
+ + + + + += =
+ +
+ + + + + += = =
+ +
+ + += =
+
3 0 0 0 3 3
0 0 0
+ + += = +
+
:
( )( )
( )( )
( ) ( )
( ) ( )
3 33
22 20 03
2 3 2 3
2 20 0
2 3
2
3 3lim lim lim
3 3 3 2 33 3
3 3 9 3 3 3 3 9 3 3lim lim
3 3 2 3 2 3
3 3 9 0 3 3 0 0 3 3 0 0 0 3 3
0 0 02 3 0 0
x
x
x
+
+ = = =
+ +
+ + = =
+
+ + = = = =
) :
( )
( )
( )
3
3
22 03
2 3 2 3
220 0
3
2
3lim lim
3 3 3
3 3 9 3 3 3 3 9 3 3lim lim
3 3 2 33 3 2 3
3 3 9 0 3 3 0 0 3 3 0 0 0 3 3
0 0 02 3 0 0
x
x
x
= =
+ + = =
+ +
+ + = = = = +
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e 2007/2008
29
:
( )
( ) ( )
( )
( )
3
3 2 3
22 20 03
2 3 2 3
2 20 0
2 3 2 3
220
3 3 3 9 3 3lim lim lim
3 3 3 2 33 3
3 3 9 3 3 3 3 9 3 3lim lim3 3 2 3 2 3
3 3 9 3 3 3 3 9 0 3 3 0 0lim
2 3 0 02 3
3 3 0 0 0 3 3
0 0 0
x
x
x
+
+ + + += = =
+ + +
+ + + + + += = =
+ + + + + += = =
+ +
+ + + = = =
+
(
7/29/2019 3967 Tok Funkcije
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e 2007/2008
30
:
( )
( )
2
2 2
2
2
2 2 2
2
2 2
5 7
5 7 5 72lim lim lim lim2 2
55 7 1
lim lim2
x x x x
x x
x xf x x x x xxk
x x x x x x
x xxx x x
x x
x x
+ + + = = = = = =
+= =
0
27
x+
0
21
x
0
1 0 01
1 0
1k
+= =
=
( )( )22
2
5 7 25 7lim lim 1 lim
2 2
lim
x x x
x
x x x xx xn f x k x x
x x
x
+ += = = =
=25 7x x +
7
2 7 3
lim lim2 2x x
x x x
x x
+ = = =
0
3
21
x
0
0 3
31 0
3n
= =
=
( )1 3y x= + , , 3y x= .
:
( )
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e 2007/2008
31
6:
:2
2
4
1
xy
=
:
, .2
2
1 0
1
1
x
x
x
, ( ) ( ) ( ), 1 1,1 1,fD = +
:
( )
( ) ( )
( )( )
22 2 2
22 221 0 0 0
2
2 2
2 220 0
1 44 1 2 4 3 2lim lim lim lim
1 1 1 21 1 21 1
3 3 3 0 0 3lim lim2 2 0 0 02
x
x
x
+ + + += = = =
+ +
= = = = = ++ + +
( )
( ) ( )
22 2 2
22 221 0 0
2
20
1 44 1 2 4 3 2lim lim lim
1 1 1 21 1 21 1
3 2 3 0 0 3lim
2 0 0 0
x
x
x
+
+ + += = = =
+ + +
+ + = = = =
( )
( ) ( )
22 2 2
22 221 0 0 0
2 2
2 20
1 44 1 2 4 3 2lim lim lim lim1 1 1 21 1 21 1
3 3 0 0 3lim
2 2 0 0 0
x
x
x
+ += = = = + +
+ + = = = =
( )
( ) ( )
( )( )
22 2 2
22 221 0 0
22 2
2 220 0 0
1 44 1 2 4 3 2lim lim lim
1 1 1 21 1 21 1
3 23 2 3 2 3 0 0 3lim lim lim
2 2 0 0 02
x
x
x
+
+ + + + += = = =
+ + +
+ + = = = = = = +
+ +
( )1x = 1x = .
7/29/2019 3967 Tok Funkcije
32/38
e 2007/2008
32
:
2
2 22 2
22
2 2
44 14
lim lim lim11x x x
x
x xx x
xx
x x
= = =
0
2
1
x
0
1 01
0 11
= =
( )1y = .
(
7/29/2019 3967 Tok Funkcije
33/38
e 2007/2008
33
8:
:2 3
2
12 3 15
9
x x xy
x
+=
:''''
( ) ( )( )
( ) ( )
2 3
2
2 3 2
2
2
12 3 150 0
9
12 3 15 0 9 0
3 4 5 0 3 3 0
3 4 5 0 3 0 3 0
x x xy
x
x x x x
x x x x x
x x x x x
+= =
+ =
+ = +
= +
( )( )
( ) ( )
( )( )
23 5 5 0 3 3
3 1 5 1 0
3 1 5 0
3 0
x x x x x x
x x x x
x x x
x
+ =
+ + =
+ =
=
( )
( )( ) ( )( )1 2 3
: 3 1 0 5 0
0 1 5 3 3
x x
x x x x x
+ = =
= = =
, 0y = : 1 0x = ( )2 1x = 3 5x = .
:
'''' - .
''''
: 0y > 0y < .
. ,
:2 3
2
12 3 150
9
x x x
x
+>
2 3
2
12 3 150
9
x x x
x
+ ( ) ( ) ( )
3, 1 0,3 5,x +
0y < ( ) ( ) ( ), 3 1,0 3,5x
?
- : 1; 0; 5 , :
( )
7/29/2019 3967 Tok Funkcije
35/38
e 2007/2008
35
'''' ( ) 0I x > ( ) 0I x <
:
: ( )I x , :
( ) ( ) ( ) ( ) ( )1 2 3 ... nI x Q x Q x Q x Q x=
: , :
( ) ( ) ( ) ( )1 2 30; 0; 0; 0;nQ x Q x Q x ... Q x= = = =
:1 2 3; ; ;
mx x x ... x (
)
: : ( ) ( ) ( ) ( ) ( )1 2 3; ; ; ... ;nQ x Q x Q x Q x I x
.
: :1 2 3
; ; ; ; ;mx x x ... x +
(1 2 3
...m
x x x x< < < < )
:
- 1
2
3
m+
( )1Q x ...
( )2Q x ...
( )3Q x ...
... ... ... ... ... ...
( )mQ x ...
( )I x ...
( )+ ( )
( )iQ x .
, ( )+ ( )
.
(
7/29/2019 3967 Tok Funkcije
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e 2007/2008
36
9:
:3
4
250
16
x x
x
>
:
:3
4
25
16
x
x
( )( )( )
( )( )
( ) ( )( )
23
4 2 2 2
25 5 525
16 4 4 2 2 4
x x x x xx x
x x x x x x
+= =
+ + +
, :
( ) ( )
( )( )( )25 5
02 2 4
x x x
x x x
+>
+ +
, :
( )
( )
1
2
3
4
5
2
0 05 0 5
5 0 5
2 0 2
2 0 2
4 0
x xx x
x x
x x
x x
x ( )
= = = =
+ = =
= =
+ = =
+
:
- 5 2 0 2 5 +
+
+
+
5 + + + + +
5 + + + + + +
2 + +
2x + + + + +
2 4x + + + + + + +
( )I x + + +
, 3
4
250
16
x x
x
>
:
( ) ( ) ( ), 5 2,0 2,5x
( )
7/29/2019 3967 Tok Funkcije
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e 2007/2008
37
10:
:2
2
2 33
4 3
x x
x x
+>
+
:
: ( ) 0I x >
( )
2
2
2
2
2 2
2
2 2
2
2
2
2 33
4 3
2 33 0
4 3
2 3 3 4 30
4 3
2 3 3 12 90
4 3
4 14 12 04 3
x x
x x
x x
x x
x x x x
x x
x x x x
x x
x xx x
+>
+
++ >
+
+ + +>
+
+ + +>
+
+> +
:
( )
( )( ) ( )
( ) ( )
( ) ( )
( ) ( )( ) ( )
2
2
2
2 2 7 60
3 3
2 2 3 4 60
1 3 1
2 2 3 2 2 30
1 3
2 2 3 20
1 3
x x
x x x
x x x
x x x
x x x
x x
x x
x x
+>
+
+>
>
>
( )( )
( )( )
: 2
2 3 20
1 3
x x
x x
>
( ):
32 3 0
2
2 0 2
1 0 1
3 0 3
x x
x x
x x
x
> >
> >
> >
> >
( )+
:
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e 2007/2008
:
- 13
22 3 +
2 3 + + +
2 + +
1 + + + +
3 +
( )I x + + +
:
( ) ( )3
,1 ,2 3,2
x
+
( )
