Mean value theorem (divided differences)

In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.^[1]

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 $f[x_0,\dots,x_n](x-x_{n-1})\dots(x-x_1)(x-x_0)$.

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 $\xi$. 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.

References

1. de Boor, C. (2005). "Divided differences". Surv. Approx. Theory 1: 46–69. MR 2221566.