(a)

The function is .

Find the critical numbers by equating the first derivative to .

Apply derivative on each side with respect to .

Equate the derivative to .

So the function has critical number at .

(b)

The critical point is , consider the table summarizes the testing of two intervals determined by the critical number.

Test interval
Test value
Sign of
Conclusion	Increasing	\  Decreasing \

The function  is increasing on the interval  and decreasing on the interval .

(c)

Use first derivative test to identify all relative extrema.

changes from positive to negative . [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 :

Graph the function is .

Observe the graph :

The function has critical number at .

The function  is increasing on the interval  and decreasing on the interval .

The function  has relative maximum at .

(a) The function has critical number at .

(b) The function  is increasing on the interval  and decreasing on the interval .

(c) The function  has relative maximum at .

(d) Graph of the function .

.