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 .