Step 1 :

Rolles Theorem :

Let f be a function that satisfies the following three hypotheses.

1. f is continuous on .

2. f is differentiable on .

3. .

Then there is a number c in such that .

Step 2 :

The function is and the interval .

The function is always continuous on the interval , because it is a polynomial.

Differentiate with respect to x.

is not differentiable at , but it is differentiable on the open interval , which satisfies the Rolles Theorem.

Check at the end points :

At :

At :

, hence Rolls theorem is applicable on .

Step 3 :

Find the value of c, such that .

The number .

Solution :

The number .