Rolle's Theorem - Generalization To Higher Derivatives

Generalization To Higher Derivatives

We can also generalize Rolle's theorem by requiring that f has more points with equal values and greater regularity. Specifically, suppose that

• the function f is n − 1 times continuously differentiable on the closed interval and the nth derivative exists on the open interval (a,b), and
• there are n intervals given by a₁ < b₁ ≤ a₂ < b₂ ≤ . . .≤ a_n < b_n in such that f(a_k) = f(b_k) for every k from 1 to n.

Then there is a number c in (a,b) such that the nth derivative of f at c is zero.

Of course, the requirements concerning the nth derivative of f can be weakened as in the generalization above, giving the corresponding (possibly weaker) assertions for the right- and left-hand limits defined above with f (n−1) in place of f.