Mean value theorem (divided differences)

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

Statement of the theorem[edit]

For any n + 1 pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point

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

For n = 1, that is two function points, one obtains the simple mean value theorem.

Proof[edit]

Let be the Lagrange interpolation polynomial for f at x0, ..., xn. Then it follows from the Newton form of that the highest term of is .

Let be the remainder of the interpolation, defined by . Then has zeros: x0, ..., xn. By applying Rolle's theorem first to , then to , and so on until , we find that has a zero . This means that

,

Applications[edit]

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

References[edit]

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