The second derivative is always positive (where defined), so
the function is concave up for all
x
±
0.
Graphical support:
[
2
4, 4] by [
2
1, 5]
(a)
[
2
1, 0) and [1,
‘
)
(b)
(
2‘
,
2
1] and (0, 1]
(c)
(
2‘
, 0) and (0,
‘
)
(d)
None
(e)
Local (and absolute) minima at (1,
e
) and (
2
1,
e
)
(f)
None
4.
Note that the domain of the function is [
2
2, 2].
y
95
x
1
}
2
ˇ
4
w
1
2
w
x
w
2
w
}
2
(
2
2
x
)
1
(
ˇ
4
w
2
w
x
w
2
w
)(1)
5
}
2
x
2
ˇ
1
4
w
(
2
w
4
2
x
w
2
w
x
2
)
}
5
}
ˇ
4
4
w
2
2
w
2
x
x
w
2
2
w
}
y
05
5
Note that the values
x
56
ˇ
6
w
are not zeros of
y
0
because
they fall outside of the domain.
Graphical support:
[
2
2.35, 2.35] by [
2
3.5, 3.5]
(a)
[
2
ˇ
2
w
,
ˇ
2
w
]
(b)
[
2
2,
2
ˇ
2
w
] and [
ˇ
2
w
,2]
(c)
(
2
2, 0)
(d)
(0, 2)
(e)
Local maxima: (
2
2, 0), (
ˇ
2
w
,2)
Local minima: (2, 0), (
2
ˇ
2
w
,
2
2)
Note that the extrema at
x
ˇ
2
w
are also absolute
extrema.
(f)
(0, 0)
5.
y
1
2
2
x
2
4
x
3
Using grapher techniques, the zero of
y
9
is
x
<
0.385.
y
052
2
2
12
x
2
52
2(1
1
6
x
2
)
The second derivative is always negative so the function is
concave down for all
x
.
Graphical support:
[
2
4, 4] by [
2
4, 2]
(a)
Approximately (
2‘
, 0.385]
(b)
Approximately [0.385,
‘
)
(c)
None
(d)
(
2‘
,
‘
)
(e)
Local (and absolute) maximum at
<
(0.385, 1.215)
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN
 Derivative

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