Rolle’s theorem says that if a function is continuous on a closed interval [*a, b*], differentiable on the open interval (*a, b*) and if *f *(*a*) = *f *(*b*), then there exists a number *c* in the open interval (*a, b*) such that . (“There exists a number” means that there is at least one such number; there may be more than one.)

The proof has two cases:

Case I: The function is constant (all of the values of the function are the same as *f *(*a*) and *f *(*b*)). The derivative of a constant is zero so any (every, all) value(s) in the open interval qualifies as *c*.

Case II: If the function is not constant then it must have a maximum or minimum in the open interval (*a, b*) by the Extreme Value Theorem. So, by Fermat’s theorem (see this post) the derivative at that point must be zero.

So, Fermat’s theorem makes Rolle’s theorem a piece of cake.

A *lemma* is a theorem whose result is used in the next theorem and makes it easier to prove. So Fermat’s theorem is a lemma for Rolle’s theorem.

On the other hand, a *corollary* is a theorem is a result (theorem) that follows easily from the previous theorem. So, Rolle’s theorem could also be called a corollary of Fremat’s theorem.

Rolle’s theorem makes a major appearance in the MVT and then more or less disappears from the stage. When you find critical number or critical points you are using Fermat’s theorem.

I like this proof because it’s so simple. It really just comes immediately from Fermat’s theorem.

The next post: The Mean Value Theorem.