help? [derivatives - shape of?]

Question

Answer 1

(10)

Step 1:

Thu function is and the interval is .

Differentiate on each side with respect to .

.

Differentiate on each side with respect to .

.

Equate to zero.

.

General solution for trigonometric equation is .

.

For ,

For ,

For ,

is not in the .

Comments

Continued....

Step 2: 

Consider the test intervals as ,, ,  and .

The graph of the function is concave up on the interval and .

The graph of the function is concave down on the interval , and .

Solution :

The function is concave up on the interval and .
Concave down on the interval , and .

Answer 2

11)

Step 1:

Thu function is .

Find the inflection points.

Inflection Point :

Inflection point is a point on the curve at which the function changes from concave up to down or vice versa.

From (10):

The curve changes concave down to concave up at and .

The curve changes concave up to concave down at and .

.

Substitute in the function.

Inflection point is .

Comments

Continued....

Step 2:

.

Substitute in the function.

Inflection point is .

Substitute in the function.

Inflection point is .

Substitute  in the function. 

Inflection point is .

Inflection points are , , and .

Solution:

Inflection points are , , and .