Math Help - Rolle's Theorem

This section of Math Help online covers Rolle's Theorem.

What is Rolle's Theorem?

Rolle's Theorem is the elementary result in mathematical analysis from which the mean value theorem follows that if a real function is continuous at and between and differentiable between, two points for which it has the same value, there is some intermediate point at which its derivative is zero.

Rolle's theorem is named after the French math analyst, algebraist, and geometer, Michael Rolle (1652-1719).

Rolle's Theorem

Let the function   f    be

• defined and continuous on the closed interval [a,b] and

• differentiable in the open interval (a,b).

Also, let

• f (a) = f (b) = 0.

Then there is at least one number   c   between  a  and  b  where    f ' (x) = 0. In another word,

f ' (c) = 0