216
CHAPTER 5 GRAPHING AND OPTIMIZATION
5
. FIRST DERIVATIVE TEST FOR LOCAL EXTREMA
Let
c
be a critical value of
f
[
(
) is defined and either
'(
) = 0 or
'(
) is not defined.]
Construct a sign chart for
'(
x
) close to and on either side of
.
f
(
c
) is a local minimum.
If
f'
(
x
) changes from negative
to positive at
c
, then
f
(
c
) is a
local minimum.
f
(
c
) is a local maximum.
If
f'
(
x
) changes from positive
to negative at
c
, then
f
(
c
) is a
local maximum.
f
(
c
) is not a local extremum.
If
f'
(
x
) does not change sign at
c
,
then
f
(
c
) is neither a local
maximum nor a local minimum.
f
(
c
) is not a local extremum.
If
f'
(
x
) does not change sign at
c
,
then
f
(
c
) is neither a local
maximum nor a local minimum.
Sign Chart
f
(
c
)


 +
+
+
f'
(
x
)
f
(
x
) Decreasing Increasing
x
m c n
)
(
+
+
+ 


f'
(
x
)
f
(
x
)
Increasing Decreasing
x
m c n
)
(


 


f'
(
x
)
f
(
x
) Decreasing Decreasing
x
m c n
)
(
+
+
+ +
+
+
f'
(
x
)
f
(
x
)
Increasing Increasing
x
m c n
)
(
6
. INTERCEPTS AND LOCAL EXTREMA FOR POLYNOMIAL FUNCTIONS
If
(
) =
a
n
+
1
+ … +
1
+
0
,
#
0 is an
th degree
polynomial then
has at most
intercepts and at most
1
local extrema.
1.
(
a, b
), (
d, f
), (
g, h
)
3.
(
b, c
), (
c, d
), (
f, g
)
5.
=
c, d, f
7.
=
b, f
9.
has a local maximum at
=
, and a local minimum at
=
;
does
not have a local extremum at
=
b
or at
=
d
.
11.
e
13.
15.
17.
19.
(
) = 2
2
 4
; domain of
: (
$
,
$
)
'(
) = 4
 4;
' is continuous for all
.
'(
) = 4
 4 = 0
= 1
Thus,
= 1 is a partition number for
', and since 1 is in the
domain of
,
= 1 is a critical value of
.