Mean value theorem (divided differences) explained

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,...,x} _{n\},max\{x} _{0,...,x} _{n\})}

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

f[x _{0,...,x} _{n]} = (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,...,x} ^{n} ^{n]x} .

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 that0 = g ^{(n)} (\xi) = f ^{(n)} (\xi) - f[x _{0,...,x} _{n]} n! ,f[x _{0,...,x} _{n]} = (f ^{(n)} (\xi))/(n!) .

Applications

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

Notes and References

1. de Boor. C.. Divided differences. Surv. Approx. Theory. 2005. 1. 46 - 69. Carl R. de Boor. 2221566.