Relative Extrema
Let
be a function defined at
.
1.
is a relative maximum of
when there exists an interval
containing
such that
for all
in
.
2.
is a relative minimum of
when there exists an interval
containing
such that
for all
in
.
If
has a relative minimum or relative maximum at
, then
is a critical number of
. That is, either
or
is undefined.
Relative extrema must occur at critical numbers as shown in picture below.
The First-Derivative Test
Let
be continuous on the interval
in which
is the only critical number, then
On the interval
, if
is negative to the left of
and positive to the right of
, then
is a relative minimum.
On the interval
, if
is positive to the left of
and negative to the right of
, then
is a relative maximum.
Guidelines for Finding Relative Extrema
1. Find the derivative of
2. Find all critical numbers, then determine the test intervals
3. Determine the sign of
at an arbitrary number in each test intervals
4. Apply the first derivative test
Exercises: Find all relative extrema of the functions below
1)
Solution:
|
Step 1: ,
|
Step 2: Critical number is , so the test intervals are and
|
Step 3: Choose for the interval , and for the interval .
|
Then we have: and
|
Step 4: is negative to the left of and positive to the right of , then is a relative minimum
|
Therefore, Relative minimum:
|
(Note: in this case is a parabola so our answer makes sense)
|
2)
Solution:
|
Step 1: ,
|
Step 2: Critical number is and , so the test intervals are and
|
Step 3: Choose for the interval , for the interval and for the interval .
|
Then we have: and
|
Step 4: is negative to the left of and positive to the right of , then is a relative minimum
|
Absolute Extrema
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