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

ξ \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)} (ξ)}{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 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, ..., 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)} (ξ) = f^{(n)} (ξ) - f [x_{0}, \dots, x_{n}] n!

,

f [x_{0}, \dots, x_{n}] = \frac{f^{(n)} (ξ)}{n!} .

Applications

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

References

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

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