Mean value theorem (divided differences)

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 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 \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

References

1. de Boor, C.. (2005). "Divided differences". Surv. Approx. Theory.