-
7/26/2019 Zadaci M
1/71
.D K . x D, ()
yK,
DK.
KDf : . yx
)(xfy= , Dx (!), Ky
( Dx )." D ! ()
f, K. # $ f%
fD
. "{ } KDxxfVf = &)( !
(!) f. '
$ !
.
RfDf: RgDg: . !
) * )()()( xgxfxh =
)()()( xgxfxh =
gfh DDD = +
) )(
)()(
xg
xfxh =
gfh DDD =
{ }-)(& = xgxDD gg . $
:
)+= -,)( Rrxxf
r
,-
R= rxxf r,)( ./ , (-,) Df.01 $
2 $ !, 23, 2 3
45,... !R.=fD
2 3 6 , Df 3 7-,).
) ,-,)( >= aaxf x R=fD +
5) ,-,89)( >= aaxxf a , ),-( +=fD +
;) xxf )( = ,R
=fD + xxf ?9
-
7/26/2019 Zadaci M
2/71
@) xxf A)(
= ,
Z
+= kkDf ,
\R
+xxf ?A)(
= ,{ } Z= kkDf ,\R +
B) xxf C2?)( = , D,7=fD + xxf C2??9
-
7/26/2019 Zadaci M
3/71
.xxf = )(
.
. ' Rx - x , ! x . I D,(=fD .
$ :
!.5 KE)( += xxf
.
.R=fD .
".( )5)( ++= xxxf .
.R
=fD
.
#.
-)( = xxf .
.L% -- >x Rx , R=fD
.
$.xe
xxf
)(
+=
.
. ' M-NJRx . I
M-NJR=fD . $
:
%. )8>()( += xxf .
5
-
7/26/2019 Zadaci M
4/71
. ' Rx
->+x , ! >x . I ),(
+=fD
. $
:
1&.
)( xxf = .. ' Rx - x . O
-5 -4 -3 -2 -1 1 2 3 4 5
-5
-4
-3
-2
-1
1
2
3
4
5
+ + - -- -
! D,7- xx. I
D,7=fD.
$ :
11. ;
@8>)(
xxxf
=
.
;
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7/26/2019 Zadaci M
5/71
. ' Rx
-;
@8>
xx, !
;
@
xx. O!
-;@;
@
+
xxxx
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-1 1 2 3 4 5 6 7 8
-10
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-8
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1
2
3
4
+ + - -- -
! D;,7-;@ + xxx . I D;,7=fD .
$ :
12.)( xxf = .
. L%-)( (
5C2?)( x
xxf +=
.
@
-
7/26/2019 Zadaci M
6/71
.
5C2?)(
=
xxf
);8>()( xxf =. !
fff DDD =
. I , $ !
$ f
f
, .
' f
! :
@5
5
xxx
. I
D@,7=fD
. f
D
:
' f ! ;-; xx . I );,(
=fD . fD
:
I , fD
$
! $ .
I );,7);,(D@,7
=== fff DDD .
1. )8>(8>)C2?()( xxxf += .
. )8>(8>)(),C2?()(
xxfxxf ==. !
fff DDD =
.
B
-
7/26/2019 Zadaci M
7/71
' f ! :
-- xxx . I D,-7=fD . fD
:
' f
! -8> >> xx . I ),(
+=fD . fD
:
fD
$
! $ .
I D,(),(D,-7
=== fff DDD .
1!.( )xxf 5C2?)( +=
.. L% 555 +
xx
, -5 >x
Rx ,
$, P=fD .
1".
;)(
+
= xx
xxf
.
.
)(,;
)(
=
= xxfx
xxf
. !
fff DDD =
.
E
-
7/26/2019 Zadaci M
8/71
' f ! -; x , ! x
x . I : ),(),(),(N,MJ
==RfD .
fD
:
' f ! - x . O
-4 -3 -2 -1 1 2 3 4
-2
-1
1
2
3
4
5
6
7
8
- - - -+ + + +
),7D,( +=fD . fD :
fD $ ! $
.
K
-
7/26/2019 Zadaci M
9/71
I
D,(),7D,(),(
+== fff DDD .
1#.
5;
;8>)( xxx
xxf +
+
=.
.
5;)(,
;8>)( xxxf
xxxf =+= .
' f ! :
+
+
,
;,-
;8> x
x
xx
x
x
,-
B
+
xx .
I , f ! ),( < xx , ),(
=fD . fD
:
' f ! -5; xx . O
H
-
7/26/2019 Zadaci M
10/71
-6 -5 -4 -3 -2 -1 1 2 3 4
-10
-8-6
-4
-2
2
4
6
8
10
+ + + +- - - -
! D,;7
=fD . fD
:
fD $ ! $
.
I ),;7
== fff DDD .
1$.
B
58>)( xx
xxf +
+
=.
.
B)(,
58>)( xxfx
xxf =
+=
.
-
-
7/26/2019 Zadaci M
11/71
' f ! :
>+
,-,-5-,-5,-
5 x
xxxxx
xx
>
,5
,5
xx
xx
.
I , f
! x
),5(),( + x,
),5(),(
+=fD .
fD :
' f ! -B x . O
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-10
-8
-6
-4
-2
2
4
6
8
10
12
14
16
18
+ + + +- - - -
-
7/26/2019 Zadaci M
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! D;,;7
=fD . ),5(),( +=fD .
fD :
fD $
! $ .
I D;,5(),;7 == fff DDD . 1%.
)8>()( xxxf = .. ' Rx
- >xx .
>
-
-
-
-
-
x
x
x
x
xx
.
O
-
7/26/2019 Zadaci M
13/71
3
3
+ + + ++ +- -- -- -
0
! :
( ) '+> kkkxx ,)(,- ( ) '< kkkxx ,,)(- .
Q >> -
-x
x
:
Q
= f
x Daaaxf ,,-,)(, K.
=
-
7/26/2019 Zadaci M
16/71
x
y
x
y
H.,,-,89)( >= aaxxf a -. ,-,89)(
-
7/26/2019 Zadaci M
17/71
x
y
x
y
5. ,A)( xxf = ;. ,?A)( xxf = =fD N,4MJ k+ =fD NMJ k ,
k , =fV k , =fV
x
y
x
y
@.D,,7,C2?)( == fDxxf B. D,,7,C2??9
-
7/26/2019 Zadaci M
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x
y
x
y
E.),,(,C2?A)( == fDxxf K. ),,(,C2??A)( == fDxxf
( )4,4 +=fV ),-( +=fV
x
y
x
y
! )(xfy= R* R !
.
! )(xfy=
)(xgy= .
( )(xfy =
Q )(xfy = ! $ ! ! )(xfy= .
K
-
7/26/2019 Zadaci M
19/71
! )(xfy =
!
)(xfy= x G.(
)( xfy =
Q )( xfy = ! $ !
! )(xfy= .
! )( xfy = !
)(xfy=
yG.
( S,)( += ccxfy
Q S,)( += ccxfy ! $
! c ! )(xfy= . ! S,)( += ccxfy
! $
* c y G ! )(xfy= . , ->c * y G
, -c * ! x G
, -= cxcfyQ
-),( >= cxcfy ! $ c ! )(xfy= .
! -),( >= cxcfy
! $
!* !* ! )(xfy=
yG >c - = ccxfy
H
-
7/26/2019 Zadaci M
20/71
Q -),( >= ccxfy ! $
c ! )(xfy
= . ! -),( >= ccxfy
! $
!* !* ! )(xfy= x G >c -
-
7/26/2019 Zadaci M
21/71
x
y
f( )x
f( )x
x
x
-
x
y
f( )x
f( )x
@
@x
x-
) $)
x
y
f( )x
f( )x ?9x
x-
8>
) !)
-
7/26/2019 Zadaci M
22/71
x
y
f( )x
f( )x
x
x5
5-
x
yf( )x x-A
f(x)Ax
) R)
2. )x
xf = 5)( + $) )()( xxf = +
) 5)( += xxf + !) )(
; +=xxf ..
x
y
f( )x
f( )x 5xf(x) 5
-
5x--
x
x
y
f( )x < =>
f( )x ( )
-x
x
) $)
-
7/26/2019 Zadaci M
23/71
x
y
f( )x 5x
f(x) 5
x
+
y
f( )x x;
f(x) x
x+
;
) !)
3. )( ))( += xxf
+ $)( ))( = xxf
+
)
)(x
xf =+ !)
)( xxf = ..
y
f( )x x
f(x) x
x
+)(
x
y
f( )x
f( )x
;x
x
( -!)
!;
) $)
5
-
7/26/2019 Zadaci M
24/71
y
f( )x x
f(x) x
x
!
y
f( )x x
f(x) x
x
) !)
. ) xxf )( = + $) )( x
xf =.
.
x
y
f( )x
f( )x
x
x
!
x
y
f( )x
f( )x
x
x
!
) $)
!. ) x
xxf
)(
+=
+ $) ;)(
+= xxxf ..
;
-
7/26/2019 Zadaci M
25/71
x
y
f( )x
f( ) x
x
x
!
+!!!!x
f(
x) x
x
y
f( )x
f( ) x
x
x
f(x) x
-( )-
( -)
) $)
( ) )( = xxf
". ) xxf )( =
+ $)xxf )( =
+
)5)( xxf = + !) 5)(
= xxxf .
.
x
y
f( )x
f( ) x
x
"x"
x
y
f( )x
f( ) x
x
"
x
"
) >
-
7/26/2019 Zadaci M
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x
f( ) x
f( )x
5
x
y
""x5
x
f( ) x
f( )x
x
y
""
x- -5
5-x-x
!) )
#. ! !
:
))( += xexf $)
)(+
=x
xxf
:fR
).,-( + :f ( ) ),-(,- + ..
x
y
f( )x
f( )y x
G
y
yx
x
f ( )x
f( )y x
G
y
yx
y
B
-
7/26/2019 Zadaci M
27/71
C) $)
$. ! !
+=
B5; xy
U
.
, 1. Q !
xy = .
x
xf( )x
y
, 2. Q !
xy 5=
! xy = %
5. $ ! xy 5=
5
, $ y G !
xy = .
x
xf( )x
y
5
E
-
7/26/2019 Zadaci M
28/71
, 3. Q !
+=
+= K5B55
xxy % !
! xy 5= K
.
x
xf( )x
y
5( )!B
+
, . Q
+==B
5;;
xyy
! $
! !
+=
B55
xy
% ;. $ ! !
!
+=
B55
xy
,
!
+=
B55
xy
.
K
-
7/26/2019 Zadaci M
29/71
x
xf( )x
y
5( )!B
+;
# $ %
1. I ! ! ,
) tax 5?9
-
7/26/2019 Zadaci M
30/71
t -
B
;
5
x a aK
55a
;
a
K
-
y - aK
a
;
a
K
55 a
=a
$) ! ! ( )ttax =
,
( )tay ?9
-
7/26/2019 Zadaci M
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I/V/WXY QW'/V/ 'ZV// ##Y/ IO O[W/OOW#//
1. I ! ! :
)
a=
$) a=
) a= !) ( ) =a.
) ' $
a=
.
- 5
;5
5 5 ;
5 5; 5
5
;
I * * , =a $
5
-
7/26/2019 Zadaci M
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$) Q a= a .
=a
) I ! a= ,
$
!.
=a
-
-
Q xay =
!) I ! ( ) =a
,
. $ * $
\ !.
5
-
7/26/2019 Zadaci M
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=a
Q ! -x
( )rxrx + -- , , ->r .
' $ -x
U
. W -y
-x , ] -, ] -,
-
7/26/2019 Zadaci M
34/71
.
W
-y
-x
, ] -, ] -, ),( -- xxx
-
7/26/2019 Zadaci M
35/71
, !* .
$ . I $ ( )= ,
( ) ( )
-
7/26/2019 Zadaci M
36/71
$ . $ $
=< -- xx
-
7/26/2019 Zadaci M
37/71
3. % !
8= =
xx
x.
.
'( )xfy=
! $ A ! x
$ , $ K
( )
-
7/26/2019 Zadaci M
38/71
Q( ) =
+xf
x8=
$ M
$ K ( ) Mxf > ! Kx> .
$ M . Kx>
( ) MKxx
x
x
xxf =>>
+=
+=
,
! MK = .
$ M $
MK = , ( ) Mxf > ! Kx> .
I
=+
+
x
x
x
8=
.
%
1. ! !
)( )8=
++
xx
x $) @
8=
E +
x
x
x
)
( )xxx
;
58=
!) H
8=
xx
)xx
x
8=
R)xxx
x+
8=
.
) Q $ ++xx ,
x , x , $ ! , .
( )8=
++
xx
x8=8=8=
++=
xxxxx
E =++= .
5K
-
7/26/2019 Zadaci M
39/71
$) Q @
+
x
x
x @ +x , ! , ! $. I
@
8=
E +
x
x
x
( )
( )@8=
8=
E
E
+
=
x
x
x
x
@8=8=
8=8=
EE
E
E
+
=
xx
xx
x
x
H
E=
=
.
) Q xx ;
5 , ;
5x x ! .
( ) ;B58=8=558= ;
;
;
===
xxxx
xxx .
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, H
x -. I-
H
8=
=
+
=
xx
.
) Q xx
! x x . ! x ! R, !
x ! . I x
x ! .
=
xxx
8= =xx
xx8=8= ( ) +=++=+
.
R) ! \
=+
xxxx
8= =+
xxxxx
8=8=
=+xxx
xxx8=8=8= ( ) ( )( ) ( ) ( ) =+=+
.
5H
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7/26/2019 Zadaci M
40/71
QW/V O# #WOLOGWV/O[/ 'ZV//
2. ! !
) 5
58=
5
x
xx
x $) xx
xx
x +
5
8=
)
B@8=
+
x
xx
x!)
;8=
x
x
x
.
) Q $
( ) -55558=
5==
xx
x ( ) -5558=
5==
x
x .
! $ $ -
-
. F
! $ ,
$ !
( ) ( )55 += xxxx . $ !
5
x
. * 5
x
$%5
x
.
=
5
58=
5 x
xx
x
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+
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5 x
xx
x( ) ;58=
5=+=+
x
x .
$) / ! $ .
Q $ ,
( ) =+ xxx ( ) ( ) ( )5 +== xxxxxxx , !
x .
xx
xx
x +
5
8= =
=
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+
8=
xxx
x
x ( )=
+
8=
xx
x
x ( ) -
=
+
.
! ) $) $
)
( )( )( ) 558=
58=
B@8=
===
=+
xx
xx
x
xx
xxx .
;-
-
7/26/2019 Zadaci M
41/71
!)
( ) ( )( ) ;8=
8=
;8=
=+=+=
+=
x
x
xx
x
x
xxx
.
3. ! !
)( )xx
x
58=$) BH5
-@@8=
++
+ xx
xx
x
) BH5
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5
;
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xx
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5558=
@
5
++
+ x
xx
x
.
) ! x 5x , $ $( ) ( ) . I * !
5x , $
( ) ( ) ( ) ==
=
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5
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xx
x
xxxx
xxx.
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! $ ++
. ! .
=++
+ BH5
-@@8=
xx
xx
x
+
+
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8=
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+=
+
xx
xxx
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) ! x $ , + . !
5x , $%
! $,;x , !
,5x ,
;
-
7/26/2019 Zadaci M
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=++
BH5
-@@8=
5
;
xx
xx
x
=+
+
555
5
555
;
BH5
-@@
8=
xx
x
x
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+
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8=
x
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5
==+
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+x .
. ! !
) 5
8=
+
++ x
x
x $) xx
xx
x +
+
58=
;
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7/26/2019 Zadaci M
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) @
E@58=
5;
;
+
++
xx
xxx
x
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5@8=
5
+
+
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B8=
5
5;
+
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xx
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)
5
5
8=5
8=5
8= =
+
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+
+
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x
x
x
xx
x
x
x
xxx
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5
8=
5
8=
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+
+
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x
xx
x
xxx
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+
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5;
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xx
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7/26/2019 Zadaci M
44/71
)
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8=5
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7/26/2019 Zadaci M
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QW/V O# /WV/O[/ 'ZV//
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x
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