help? derivatives?

Question

Answer 1

6)

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 points.

Thus, the critical points exist when .

Equate  to zero.

, and .

The critical points are , and .

The test intervals are , , and .

The function is increasing on the intervals and .

The function is decreasing on the intervals and .

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Contd..

Step 2:

.

Differentiate on each side with respect to .

Find the inflection points.

Equate to zero.

The inflection points are at .

The test intervals are , and .

The graph is concave up on the interval and .

The graph is concave down on the interval .

Step 3:

Graph the function :

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 .