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, ..., x_n in the domain of an n-times differentiable function f there exists an interior point

where the nth derivative of f equals n&nbsp;<nowiki>!</nowiki> times the nth divided difference at these points:

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

Proof

Let be the Lagrange interpolation polynomial for f at x₀,&nbsp;...,&nbsp;x_n. Then it follows from the Newton form of that the highest order term of is .

Let be the remainder of the interpolation, defined by . Then has zeros: x₀,&nbsp;...,&nbsp;x_n. 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.

References