(a)

The cost function is .

Find .

Substitute  in the cost function.

Rewrite the expression.

Find .

Substitute  in the cost function.

Therefore, .

(b)

Rolles Theorem :

Let  be a function that satisfies the following three hypotheses.

1.   is continuous on .

2.   is differentiable on .

3. .

Then there is a number  in  such that .

 is continuous on  and differentiable on the interval 

.

So the function  satisfies the three hypotheses.

Therefore there is a number  in  such that .

The cost function is .

Apply derivative on each side with respect to .

Equate it to zero. \ \

Hence rate of change of the cost function is zero at .

(a) .

(b) Rate of change of the cost function is zero at .