# Rolle's Theorem

**Rolle's theorem** is an important theorem among the class of results regarding the value of the derivative on an interval.

## Statement

Let

Let be continous on and differentiable on

Let

Then such that

## Proof

The result is trivial for the case . Hence, let us assume that is a non-constant function.

Let and Without loss of generality, we can assume that

By the Maximum-minimum theorem, such that

Assume if possible

Let

Hence, such that

i.e. ,

Thus we have that if , contradicting the assumption that is a maximum.

Similarly we can show that leads to contradiction.

Therefore,

QED