$\"\"$

The function is $\"\"$ , where $\"\"$ is a positive constant.

Graph :

Graph the function $\"\"$ :

$\"\"$

The function is $\"\"$ .

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

$\"\"$

Find the critical points.

The critical points exist when $\"\"$ .

Equate $\"\"$ to zero.

$\"\"$

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

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

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

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

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

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

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

$\"\"$

Find the local maximum and local minimum.

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

Substitute $\"\"$ in the function.

$\"\"$

Local maximum is $\"\"$ .

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

Substitute $\"\"$ in the function.

$\"\"$

Local minimum is $\"\"$ .

$\"\"$

$\"\"$ .

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

$\"\"$

Find the inflection points.

Equate $\"\"$ to zero.

$\"\"$

The inflection point is $\"\"$ .

Substitute $\"\"$ in the function.

$\"\"$

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 $\"\"$ .

Observe the above tests.

If $\"\"$ is any positive number, the function does not change the increasing and decreasing intervals and concavity also does not change.

But local maximum and local minimum as well as inflection points varies.

$\"\"$

The increasing and decreasing intervals and the concavity are same.

The local maximum and local minimum as well as inflection points differs.