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 ! 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, ..., 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, ..., x_n.
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.