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# Step-by-Step Math Answer

Find the derivative of the function
*
g
*
(
*
x
*
).

To find the relative extrema of the function, first find the derivative of the function.

*
g
*
′ (
*
x
*
) = sech
*
x
*
–
*
x
*
sech
*
x
*
tanh
*
x
*
or sech
*
x
*
(1 –
*
x
*
tanh
*
x
*
)

Set
*
g
*
′ (
*
x
*
) equal to 0.

Next, set
*
g
*
′ (
*
x
*
) equal to 0.

sech
*
x
*
(1 –
*
x
*
tanh
*
x
*
) = 0

Simplify.

*
x
*
tanh
*
x
*
= 1

Use the definition of the Hyperbolic Functions.

By the definition of the Hyperbolic Functions

.

So,

.

Rewrite as exponential functions and simplify.

Now, rewrite as exponential functions.

Simplify.

*
e
^{
x
}
*
(

*x*– 1) –

*e*(

^{ –x }*x*+ 1) = 0

Use the
**
solver
**
function in the

**option in a graphing calculator.**

*math*
To solve for
*
x
*
, use the
**
solver
**
function in the

**option in a graphing calculator.**

*math*
*
x
*
± 1.20

Make a table that shows the testing of three intervals determined by the values of
*
x
*
.

Next, make a table that shows the testing of three intervals determined by the values of
*
x
*
.

So,
*
f
*
is decreasing on the intervals (–
, –1.20) and (1.20,
) and increasing on the interval (–1.20, 1.20).

A relative minimum occurs where
*
g
*
changes from decreasing to increasing.

A relative minimum occurs where
*
g
*
changes from decreasing to increasing.

From the table, the function changes from decreasing to increasing at the point where
*
x
*
= –1.20.

So, the function has a relative minimum at the point where
*
x
*
= –1.20.

Find
*
g
*
(–1.20).

If
*
g
*
′ (
*
x
*
) changes from negative to positive at
*
c
*
, then
*
g
*
has a relative minimum at (
*
c
*
,
*
g
*
(
*
c
*
)).

To find the relative minimum, find
*
g
*
(–1.20).

*
g
*
(–1.20) = –1.20 sech (–1.20) or –0.7

So, the function has a relative minimum at the point (–1.20, –0.7).

A relative maximum occurs where
*
f
*
changes from increasing to decreasing.

A relative maximum occurs where
*
f
*
changes from increasing to decreasing.

From the table, the function changes from increasing to decreasing at the point where
*
x
*
= 1.20.

So, the function has a relative maximum at the point where
*
x
*
= 1.20.

Find
*
g
*
(1.20).

If
*
g
*
′ (
*
x
*
) changes from positive to negative at
*
c
*
, then
*
f
*
has a relative maximum at (
*
c
*
,
*
f
*
(
*
c
*
)).

To find the relative maximum, find
*
g
*
(1.20).

*
g
*
(1.20) = 1.20 sech (1.20) or 0.7

So, the function has a relative maximum at the point (1.20, 0.7).

Use the
**
graph
**
feature in a graphing calculator.

To graph the function, first enter the given function in a graphing calculator.

Next, use the
**
graph
**
feature.

In our case, the graph obtained is as shown.

Observe the graph.

Observe the graph.

The function
*
f
*
is increasing on the interval (–1.20, 1.20) and decreasing on the intervals (–
, –1.20), and (1.20,
).

So, the result is correct.