Consider a function which is continuous everywhere on an interval and has a second derivative everywhere on the open interval . Assume the chord joining two points and on the graph of the function intersects the graph of the function at a point with . Prove there exists a point such that .

* Proof. * Let be the equation of the line joining and . Then define a function

Since and intersect at the values , and this means

By our definition of then we have

Further, since and are continuous and differentiable on , we apply Rolle’s theorem twice: first on the interval and then on the interval . These two applications of Rolle’s theorem tells us there exist points and such that

Then, we apply Rolle’s theorem for a third time, this time to the function on the interval to conclude that there exists a such that . Then, since we know (since ). So,

Thus, for some