$\"\"$

(a)

The function is $\"\"$ .

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

$\"\"$

Find the critical points.

Since $\"\"$ is a polynomial it is continuous at all the point.

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

Equate $\"\"$ to zero:

$\"\"$

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

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

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

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

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

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

The function is increasing on the intervals $\"\"$ and $\"\"$ .

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

$\"\"$

(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 $\"\"$ .

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

$\"\"$

Local maximum is $\"\"$ .

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

$\"\"$

Local minimum is $\"\"$ .

$\"\"$

(c)

$\"\"$ .

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

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

The inflection point is $\"\"$ and $\"\"$ .

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

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

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

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

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

$\"\"$

Inflection points :

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

$\"\"$

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

$\"\"$

The inflection point is $\"\"$ .

(a)

Increasing on the intervals $\"\"$ and $\"\"$ .

Decreasing on the interval $\"\"$ and $\"\"$ .

(b)

Local maximum is $\"\"$ .

Local minimum is $\"\"$ .

(c)

Concave up on the interval $\"\"$ and $\"\"$ .

Concave down on the interval $\"\"$ .

Inflection points $\"\"$ .