$\"\"$

(a)

The function is $\"\"$ .

Find the critical numbers by applying derivative.

$\"\"$

Apply derivative on each side with respect to $\"\"$ .

$\"\"$

Equate the derivative to $\"\"$ .

$\"\"$

The function has a discontinuity at $\"\"$ and $\"\"$ .

Therefore critical number is $\"\"$ .

$\"\"$

(b)

The function has a discontinuity at $\"\"$ and $\"\"$ .

So we need consider the discontinuity while setting the test intervals.

Consider the test intervals to find the interval of increasing and decreasing.

Test intervals are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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

Test interval	Test value	Sign of $\"\"$	Conclusion
$\"\"$	$\"\"$	\ $\"\"$ \	Increasing
$\"\"$	$\"\"$	\ $\"\"$ \	Increasing
$\"\"$	$\"\"$	\ $\"\"$ \	Decreasing
$\"\"$	$\"\"$	\ $\"\"$ \	Decreasing

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

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

$\"\"$

(c)

Use first derivative test to identify all relative extrema.

$\"\"$ changes from positive to negative at $\"\"$ . [From (b) ]

Therefore according to first derivative test , the function has maximum at $\"\"$ .

The function $\"\"$ has a relative maximum at $\"\"$ .

Find $\"\"$ .

$\"\"$

So the function $\"\"$ has relative maximum at $\"\"$ .

$\"\"$

(d)

Graph the function is $\"\"$ .

$\"\"$

Now observe the graph :

Critical number is $\"\"$ .

The function has a discontinuity at $\"\"$ and $\"\"$ .

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

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

The function $\"\"$ has relative maximum at $\"\"$ .

$\"\"$

(a)

Critical number is $\"\"$ .

The function has a discontinuity at $\"\"$ and $\"\"$ .

(b)

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

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

(c)

The function $\"\"$ has relative maximum at $\"\"$ .

(d)

Graph of the function is $\"\"$ .

$\"\"$ .