(10)

Step 1:

The function is $\"\"$ .

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

$\"\"$

Find the critical points.

Since it is a polynomial it is continuous at all the point.

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

Equate $\"\"$ to zero.

$\"\"$

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

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

The test intervals are $\"\"$ .

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

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

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

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

$\"\"$ in the interval of $\"\"$ .

So the function $\"\"$ is increasing at $\"\"$ .

(b)

Find the local maximum and local minimum.

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

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

$\"\"$

Local maximum is $\"\"$ .

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

$\"\"$

Local minimum is $\"\"$ .

(c)

$\"\"$ .

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

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

The inflection point is $\"\"$ .

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

$\"\"$

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

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

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

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

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

The inflection point is $\"\"$ .

(a)

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

Decreasing on the interval $\"\"$ .

(b)

Local maximum is $\"\"$ .

Local minimum is $\"\"$

(c)

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

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

Inflection point is $\"\"$ .