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

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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

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

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

References

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