(a)

The function is . 

Graph of the function is shown.

Fundamental theorem of calculus: 

If is continuous on , then the function is defined by  is continuous on  and differentiable on , then .

Here .

Therefore,graph of the function is same as graph of .

First derivative test : 

Consider is a critical number of a continuous function . 

(i) If changes from positive to negative at , then has a local maximum at .

(ii) If changes from negative to positive at , then has a local minimum at .

Observe the graph.

have local maxima at and , because changes its sign from positive to negative.

 have local minima at and because changes its sign from negative to positive.  

(c)

Concave downward :

The function is concave down when is decreasing in the interval.

Observe the graph.

is decreasing on , and .

The function is concave downward over the interval , and . 

(d)

Rough graph of .

   \ \

The curve is concave down on the interval .