The function is a positive and differentiable on entire real line.

(a) If is increasing, must be increasing.Explain.

The function .

Apply derivative on each side with respect to .

.

The function is increasing and is positive.

.

A function is increasing when its derivative is positive.

Therefore, the function is increasing.

(b) If the graph of is concave upward, must the graph of be concave upward.Explain.

A function is concave up when its double derivative is positive.

Again apply derivative on each side with respect to .

.

The is may or may not be positive, even though .

is may or may not be concave upward if is concave upward.

Consider the function .

The function .

Graph the functions: and .

Observe the graph:

The function is concave up.

The function is concave down.

is may or may not be concave upward if is concave upward.

(a) Yes. If is increasing, must be increasing.

(b) is may or may not be concave upward if is concave upward.