and are positive, increasing and concave upwards in the interval .

Concavity test:

(a) If  for all in , then the graph of  is concave upward on .

(b) If  for all in , then the graph of is concave downward on .

From the question we can draw the following information.

1. and , since the they are positive.

2. and , since the they are increasing.

3. and , since the they are concave upwards.  [From concavity test] 

Let the product of two functions is .

Let be the number in the interval .

Apply derivative on each side with respect to .

Apply second derivative on each side with respect to .

From the above conditions , and . 

So

From the second derivative test product of two functions is also concave upwards in .

Product of two functions is also concave upwards in .