help-derivatives? shape of?

Question

Answer 1

(5)

Step 1:

The function is .

Differentiate on each side with respect to .

.

Step 2:

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 .

Solution :

The critical points are and .

Answer 2

(6)

Step 1:

The function is .

.

The critical points are and .

The test intervals are , and .

The function is increasing on the intervals and .

The function is decreasing on the interval .

Solution :

The function is increasing on the intervals and .

The function is decreasing on the interval .

Answer 3

(7)

Step 1:

The function is .

The critical points are and .

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 .

Local maximum is .

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

Local minimum is .

Solution :

Local maximum is .

Local minimum is .