Step 1:

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 $\"\"$ .

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

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

$\"\"$

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

$\"\"$

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

$\"\"$

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 $\"\"$ .

Step 2:

$\"\"$ .

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

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

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

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

$\"\"$

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

$\"\"$

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

The inflection point is $\"\"$ .

Step 3:

Graph the function $\"\"$ :

Graph :

1). Draw a coordinate plane.

2). Plot the inflection points and critical points.

3). Connect the plotted points.

$\"\"$

Solution: \ \

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

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

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

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

\ \

Graph of the function $\"\"$ :

$\"\"$ .