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 incerasing, must  be increasing.

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