The function on .

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 .

(a)

Explain why Rolles Theorem does not apply.

The function is .

In this case .

Substitute in .

.

Substitute in .

.

.

The function does not holds the rolles theorem.

(b)

Conclusion of Rolles Theorem is true for .

The function is .

Apply derivative on each side with respect to .

.

Conclusion of Rolles theorem:

There is a number  in  such that .

.

The value of lies on interval .

The conclusion of Rolles Theorem is true for .

(a) The function does not holds the rolles theorem.

(b) The conclusion of Rolles Theorem is true for .