IXL - Classify local extrema using the second derivative test (Calculus practice)

The function

(

)

–

has critical points at

–

and

Use the second derivative test to classify each critical point. If the test cannot be used, select

Inconclusive.

ref_doc_title.

Learn with an example

question

The function

(

)

–

has critical points at

–

and

Use the second derivative test to classify each critical point. If the test cannot be used, select

Inconclusive.

	Local minimum	Local maximum	Inconclusive
x = – 2
x = 2

key idea

The second derivative test states that if

has a critical point at

and

′′

(

)

exists, then the following statements are true.

If
f
′′
(
c
)
>
0
,
then
f
has a local minimum at
x
=
c
.
If
f
′′
(
c
)
<
0
,
then
f
has a local maximum at
x
=
c
.
If
f
′′
(
c
)
=
0
,
then the test is inconclusive;
f
may have a local minimum, a local maximum, or neither at
x
=
c
.

solution

Start by finding the first derivative.

(

)

–

′

(

)

(

–

)

′

–

The function

has critical points at

–

and

Find the second derivative and evaluate

′′

(

–

)

and

′′

(

)

′′

(

)

(

–

)

′

′′

(

–

)

(

–

)

–

′′

(

)

(

)

Now, use the second derivative test.

Since
f
′′
(
–
2
)
<
0
,

f
has a local maximum at
x
=
–
2
.
Since
f
′′
(
2
)
>
0
,

f
has a local minimum at
x
=
2
.

$Confetti$

Excellent!

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J.3Find second derivatives of trigonometric, exponential, and logarithmic functions

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