Prove, using Rolle’s theorem, that for any value of there is at most one such that

* Proof. * The argument is by contradiction. Suppose there are two or more points in for which . Let be two such points with . Since are both in we have

Since is continuous on and differentiable on (all polynomials are continuous and differentiable everywhere), we may apply Rolle’s theorem on the interval to conclude that there is some such that

But this contradicts that (since and ). Hence, there can be at most one point such that

Yes, but with a direct proof, I am not sure how Rolle’s Theorem applies, since we do not have end point equality.