Mean value theorem (divided differences)


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

Let be the Lagrange interpolation polynomial for f at x0, ..., xn.
Then it follows from the Newton form of that the highest term of is.
Let be the remainder of the interpolation, defined by. Then has zeros: x0, ..., xn.
By applying Rolle's theorem first to, then to, and so on until, we find that has a zero. This means that

Applications

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