Math021, week 8
Example 8.1
Find all the relative maximum and minimum of the function
f
(
x
) =
xe
x

e
x

x
2
.
solution:
Since
f
0
(
x
) =
x
(
e
x

2)
>
0
x >
ln 2 or
x <
0
= 0
x
= 0 or ln 2
<
0
0
< x <
ln 2
f
has a relative minimum at ln 2 and a relative maximum at 0.
Concavity
Definition 8.2
The graph of a function
f
is concave up (down) if
f
0
is increas
ing (decreasing).
Corollary 8.3
If
f
00
>
(
<
)0
, the graph of
f
is concave up.
Definition 8.4
f
has a point of inflection at
c
if
f
changes its concavity at
c
.
Example 8.5
Find the points of inflection of
f
(
x
) =
e

x
2
/
2
and hence sketch
its graph.
solution:
f
0
(
x
) =

xe

x
2
/
2
,
f
00
(
x
) = (
x
2

1)
e

x
2
/
2
. Thus,
f
0
(
x
) =

x
(
e
x
2
/
2
)
>
0
x <
0
= 0
x
= 0
<
0
0
< x
f
has a relative maximum at 0. Moreover,
f
00
(
x
) = (
x

1)(
x
+ 1)
e

x
2
/
2
>
0
x >
1 or
x <

1
= 0
x
= 1 or

1
<
0

1
< x <
1
Thus,
f
has points of inflection at

1 and 1. The graph of
f
is concave up on
[1
,
+
∞
) and (
∞
,

1] and it is concave down on [

1
,
1]. Finally, note that the
graph of
f
has a horizontal asymptote
y
= 0.
1
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Theorem 8.6 (Second Derivative Test)
Suppose that
c
is a critical point
of a function
f
.
1. If
f
00
(
c
)
>
0
,
f
has a relative minimum at
c
.
2. If
f
00
(
c
)
<
0
,
f
has a relative maximum at
c
.
3. If
f
00
(
c
) = 0
,
f
may have a relative maximum, minimum or neither of
them.
proof:
If
f
00
(
c
)
>
0,
f
00
(
x
)
>
0 when
x
is close to
c
.
Thus,
f
0
is increasing over an
interval including
c
. But we know that
f
0
(
c
) = 0. Therefore,
f
0
(
x
)
≤
0 when
x
is a bit less than
c
and
f
0
(
x
)
≥
0 when
x
is a bit greater than
c
.
f
has a relative
minimum at
c
. We may deal with the remaining case in which
f
00
(
c
)
<
0 in a
similar way.
Example 8.7
Find all the relative maximum and minimum of the function
f
(
x
) = 2
x
3
+ 3
x
2

12
x
+ 7
.
solution:
Since
f
0
(
x
) = 6(
x
+ 2)(
x

1).
The critical points of
f
are

2 and 1.
Now,
f
00
(
x
) = 12
x
+ 6 so that
f
00
(

2) =

18
<
0,
f
00
(1) = 18
>
0. Hence,
f
has a
relative maximum at

2 and a relative minimum at 1.
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 Fall '10
 LUXNU
 Math, Calculus, Derivative, Mathematical analysis

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