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x2x10 x
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f(x 1)
f(x 2)
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A07:69
0 x
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a b c d
f(a) m ínimoabsoluto
f(b) m áximorelativo
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f(d) m áximoabsoluto
0 x
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f(a) m ínimoabsoluto
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07 ,
07 =0 9 2HC.9 /0
0=H ,-,4A9 /, 0., ,82080 07 ;?,6;?0 ?7 /9= :989= /0
?8JI9 .97 , .98.,@4/,/0
@96,/, :,, .47,
?8JI9 .97 , .98.,@4/,/0
@96,/, :,, -,4A9
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
56. #9:40/,/0= /,= ?8JO0=
#989= /0 8D0AI9
0C84JI9
(7, 1?8JI9 : R R /4B =0 ;?0 07 ?7
! 07
- =0 9 =084/9 /, .98.,@4/,/0 G 0=;?0/, /0 0 G /404, /0 K
/410080
PontodeInflexão
Concavidade
voltadaparacima
Concavidade
voltadaparabaixo
PontodeInflexão
Concavidade
voltadaparacima
Concavidade
voltadaparabaixo
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
57. #9:40/,/0= /,= ?8JO0=
98.,@4/,/0 0 #989= /0 8D0AI9 A07:69
A07:69
-=0@0 , 0:0=08,JI9 2HC., /, 1?8JI9 48/4.,8/9 ,= .98.,@4/,/0= 0
:989= /0 48D0AI9 /0
-3 -2 -1 1 2 3 4
-6
5
11
x
y
-15
( )f x
-3 -2 -1 1 2 3 4
-6
5
11
x
y
-15
( )f x
96?JI9
*# %$, ,%* ( -% # 59. *# +# %$*% $1-4% # -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
60. #9:40/,/0= /,= ?8JO0=
?8JI9 474,/,
0C84JI9
05, : R R 0 /0=428079= :9 ?7, :,0 /9 =0? /97M849
1?8JI9 /4B =0
07 =0 0A4=07 # 0 0,4= ,6
;?0 # - :,, 9/9= 9= @,690= - :008.080= ,
#
R-
: # -
0C84JI9
(7, 1?8JI9 /4B =0
07 =0 8I9 19 6474,/, 07
?8JI9 474,/, 07
-1 0 1
1
x
y
( ) 2
f x x=
-1 0 1
1
x
y
( ) 2
f x x=
?8JI9 6474,/, 07
0 1 2 3
1
2
3
x
y
( )
1
g x
x
=
0 1 2 3
1
2
3
x
y
( )
1
g x
x
=
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
63. 0 , 1?8JI9 K :9=44@, 0 =0 , 1?8JI9 K 802,4@,
!I9 K =9-050.4@, 807 4850.4@,
F 6474,/,
0
b
x
y
y = b
b0
0
b
x
y
y = b
b0
0
b
x
y
y = b
b0
0
b
x
y
y = b
b0
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
65. R
0:0=08,JI9 2HC., K ?7, 0., 9-6M;?, .97 ?7 P84.9 B09 07
#
66. # K 9 /0.64@0 9? 48.648,JI9 /, 0., 07 06,JI9 ,9 04A9 /,= ,-.4==,=
K , 9/08,/, 8, 94207
F -450.4@, =9-050.4@, 0 4850.4@, !I9 K 6474,/,
0 # , 1?8JI9 K .0=.080 0 =0 # , 1?8JI9 K /0.0=.080
0
b
x
y
m0
0
b
x
y
m0
y mx b= + y mx b= +
b
m
−
b
m
−0
b
x
y
m0
0
b
x
y
m0
y mx b= + y mx b= +
b
m
−
b
m
−
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
67. =-9J, 9 2HC.9 /0 ?7, 1?8JI9
%0.,
;?,JI9 20,6 /, 0., - .
;?,JI9 /0 ?7, 0., ;?0 :,==, :069 :989 - . 0 07 /0.64@0 # :
. . # - -
;?,JI9 0/?B4/, 0.,= 8I9 @04.,4= /0.64@0 # 0 9/08,/, 8, 94207 :
. #-
%0., @04.,6 -
%0., 394B98,6 .
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
68. 0.64@0 .90C.4080 ,82?6, /0 ?7, 0.,
/0 ?7, 0., 8I9 @04.,6 ;?0 :,==, :9 -
.
0
- . K # . .
- -
- -
#,, 0.,= @04.,4= 9 /0.64@0 K 48/0C84/9
x
y
m=0
m0
m0
m indefinido
/0.64@0 /0 ?7, 0., :9/0 =0 480:0,/9 .979 ,BI9 04A9= .97 , 70=7,
?84/,/0 /0 70/4/, 9? .979 ,A, /0 @,4,JI9 04A9= .97 /410080= ?84/,/0= /0
70/4/,
#
@,4,JI9 @04.,6
@,4,JI9 394B98,6
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
69. %0.,= #,,606,= 0 #0:08/4.?6,0=
?,= 0.,= /4=48,= 8I9 @04.,4=
=I9
==0 #
# x
y
m1
m2m1=m2
?,= 0.,= /4=48,= 8I9 @04.,4=
=I9
==0 #
#
x
y
m1
m2
m1=-1/m2
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
?8JO0= 60708,0=
?8JI9 $?,/H4.,
0C84JI9
= 1?8JO0= /, 197, - - - 98/0
R 0
/0=428,7 =0 :9
98=4/0,8/9 ?7, 1?8JI9 ;?,/H4., - - - ,= =?,= ,MB0=
0,4= :9/07 =0 9-4/,= ,,@K= /, N7?6, %0=96@080
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
70. ?8JO0= 60708,0=
?8JI9 $?,/H4.,
(7, 1?8JI9 ;?,/H4., 07 ,= =02?480= :9:40/,/0=
R
0:0=08,JI9 2HC., K ?7, :,H-96, .?5,= .99/08,/,= /9 @K4.0
=I9
∆
98/0 ∆
71. @,69 ,-=96?9 /0 /00748, , ,-0?, /, 1?8JI9 9? =05,
;?,89 7,49 19 7,4= 10.3,/, K , :,H-96,
!I9 K =9-050.4@, 807 4850.4@, !I9 K 6474,/,
98.,@4/,/0 @96,/, :,, .47, 98.,@4/,/0 @96,/, :,, -,4A9
73. ∞ ∆
0 x
y
a
b
2
−
a4
∆
−
V
0 x
y
a
b
2
−
a4
∆
−
V
0 x
y
a
b
2
−
a4
∆
− V
0 x
y
a
b
2
−
a4
∆
− V
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
74. ?8JO0= 60708,0=
?8JI9 #9648974,6
= 1?8JO0= ,8,64=,/,= ,80497080 .98=,80 ,C7 0 ;?,/H4., =I9
.,=9= :,4.?6,0= /0 1?8JO0= :9648974,4=
0C84JI9
= 1?8JO0= 0:0=08,/,= /, =02?480 197,
- $-$
$
-$
-
/0=428,7 =0 :9
= 8P709= 0,4= $ $
. . .
=I9 /0=428,/9= :9
=0 $ , 1?8JI9 :9648974,6 /4B =0 /0
75. 98/0 $ K ?7 8P709 48049 :9=44@9
(7, 1?8JI9 :9648974,6 /0 2,? 76. /0=428, =0 :9 1?8JI9 .P-4., $?,6;?0
1?8JI9 :9648N74,6 /0 2,? $ 07 89 7HA479 $ ,MB0= 0,4= /4=48,=
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
78. R +09 07 -
!I9 K =9-050.4@, 807 4850.4@, !I9 K 6474,/,
79. 0.0=.080 07 R 0 .0=.080 07 R M8479 42?,6 , 07 -
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
( )f x x=
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
( )f x x=
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
=-9J, 9 2HC.9 /0 ?7, 1?8JI9
62?7,= ?8JO0= H=4.,=
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
?8JI9 4/084/,/0
- -
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
?8JI9 ;?,/H4.,
- -
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
?8JI9 .P-4.,
- - 80. 0 1 2 3 4
0
1
2
3
4
x
y
?8JI9 ,4B ;?,/,/,
-
-
-4 -3 -2 -1 0 1 2 3 4
1
2
3
4
x
y
?8JI9 7N/?69
- -
-4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
4
x
y
?8JI9 ,.498,6
-
-
-10 -8 -6 -4 -2 2 4 6 8 10
-1.0
-0.5
0.5
1.0
x
y
?8JI9 =089
- sen -
-10 -8 -6 -4 -2 2 4 6 8 10
-1.0
-0.5
0.5
1.0
x
y
?8JI9 .9=089
- .9= -
0: /0 ,07H4., '0P-,6 # 98.049= H=4.9= ,8
