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


 * $$ \xi \in (\min\{x_0,\dots,x_n\},\max\{x_0,\dots,x_n\}) \,$$

where the nth derivative of f equals n ! times the nth divided difference at these points:


 * $$ f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}.$$

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

Proof
Let $$P$$ be the Lagrange interpolation polynomial for f at x0, ..., xn. Then it follows from the Newton form of $$P$$ that the highest order term of $$P$$ is $$f[x_0,\dots,x_n]x^n$$.

Let $$g$$ be the remainder of the interpolation, defined by $$g = f - P$$. Then $$g$$ has $$n+1$$ zeros: x0, ..., xn. By applying Rolle's theorem first to $$g$$, then to $$g'$$, and so on until $$g^{(n-1)}$$, we find that $$g^{(n)}$$ has a zero $$\xi$$. This means that


 * $$ 0 = g^{(n)}(\xi) = f^{(n)}(\xi) - f[x_0,\dots,x_n] n!$$,
 * $$ f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}.$$

Applications
The theorem can be used to generalise the Stolarsky mean to more than two variables.