Mean value theorem (divided differences)

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The mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. It states that for given n + 1 points of the function f and the divided difference f[x_0,\dots,x_n] with respect to these points, there exists an interior point where the nth derivative of f equals the divided difference.

\exists \xi\in[\min\{x_0,\dots,x_n\},\max\{x_0,\dots,x_n\}] \ f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}

[edit] Special cases

For n = 1, that is two function points, you obtain the simple mean value theorem.

[edit] Applications

The theorem can be used to define a mean like the Stolarsky mean.

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