(a)

Thu function is $\"\"$ , $\"\"$ .

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

Find the critical points.

Thus, the critical points exist when $\"\"$ .

Equate $\"\"$ to zero:

$\"\"$

$\"\"$ and $\"\"$

$\"\"$ and $\"\"$ .

The critical points are $\"\"$ and $\"\"$ .

The test intervals are $\"\"$ and $\"\"$ .

\ \

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Interval	Test Value	Sign of $\"\"$	Conclusion
$\"\"$	$\"\"$	\ $\"\"$ \	Decreasing
$\"\"$	$\"\"$	\ $\"\"$ \	Increasing

The function is increasing on the interval $\"\"$ .

The function is decreasing on the interval $\"\"$ .

(b)

Find the local maximum and local minimum.

The function $\"\"$ has a local minimum at $\"\"$ , because $\"\"$ changes its sign from negative to positive.

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Local minimum is $\"\"$ .

(c)

$\"\"$ .

Differentiate $\"\"$ on each side with respect to $\"\"$ .

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

$\"\"$ and $\"\"$

$\"\"$ and $\"\"$ and $\"\"$ .

$\"\"$ is not considered as a infletion point since it is critical point.

The inflection points are at $\"\"$ and $\"\"$ .

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Substitute $\"\"$ in $\"\"$ .

$\"\"$

The test intervals are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

\ Interval \	Test Value	Sign of $\"\"$	Concavity
$\"\"$	$\"\"$	$\"\"$	Down
$\"\"$	$\"\"$	$\"\"$	\ Up \
$\"\"$	$\"\"$	$\"\"$	Down

The graph is concave up on the interval $\"\"$ .

The graph is concave down on the intervals $\"\"$ and $\"\"$ .

The inflection points are $\"\"$ and $\"\"$ .

(d)

Graph :

Graph the function $\"\"$ , :

$\"\"$

(a)

Increasing on the interval $\"\"$ .

Decreasing on the interval $\"\"$ .

(b)

Local minimum is $\"\"$ .

(c)

Concave up in the interval $\"\"$ .

Concave down in the intervals $\"\"$ and $\"\"$ .

Inflection points are $\"\"$ and $\"\"$ .

(d)

Graph of the function $\"\"$ is

$\"\"$ .