In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.
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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
Let
Applications
The theorem can be used to generalise the Stolarsky mean to more than two variables.