In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.

Statement of the theorem

For any n+1 pairwise distinct points x0,...,xn in the domain of an n-times differentiable function f there exists an interior point

where the nth derivative of f equals n! times the nth divided difference at these points:

For n=1, that is two function points, one obtains the simple mean value theorem.

Proof

be the Lagrange interpolation polynomial for f at x0,...,xn. Then it follows from the Newton form.

zeros: x0,...,xn. By applying Rolle's theorem. This means that

1. ,

Applications

The theorem can be used to generalise the Stolarsky mean to more than two variables.

References