Mean value theorem (divided differences)
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.
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
The theorem can be used to generalise the Stolarsky mean to more than two variables.