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

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

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, ..., x_n. Then it follows from the Newton form of P that the highest 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, ..., x_n. 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 ξ. 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.