Mean value theorem (divided differences)

In mathematical analysis, the '''mean value theorem for divided differences''' generalizes the mean value theorem to higher derivatives.<ref>{{cite journal|last=de Boor|first=C.|title=Divided differences|journal=Surv. Approx. Theory|year=2005|volume=1|pages=46–69|authorlink=Carl R. de Boor|mr=2221566}}</ref>

== Statement of the theorem ==

For any ''n'' + 1 pairwise distinct points ''x''0, ..., ''x''''n'' in the domain of an ''n''-times differentiable function ''f'' there exists an interior point

: <math> \xi \in (\min\{x_0,\dots,x_n\},\max\{x_0,\dots,x_n\}) \,</math>

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

: <math> f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}.</math>

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

== Proof ==

Let <math>P</math> be the Lagrange interpolation polynomial for ''f'' at ''x''0, ..., ''x''''n''. Then it follows from the Newton form of <math>P</math> that the highest order term of <math>P</math> is <math>f[x_0,\dots,x_n]x^n</math>.

Let <math>g</math> be the remainder of the interpolation, defined by <math>g = f - P</math>. Then <math>g</math> has <math>n+1</math> zeros: ''x''0, ..., ''x''''n''. By applying Rolle's theorem first to <math>g</math>, then to <math>g'</math>, and so on until <math>g^{(n-1)}</math>, we find that <math>g^{(n)}</math> has a zero <math>\xi</math>. This means that

: <math> 0 = g^{(n)}(\xi) = f^{(n)}(\xi) - f[x_0,\dots,x_n] n!</math>, : <math> f[x_0,\dots,x_n] = \frac{f^{(n)}(\xi)}{n!}.</math>

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

== References == {{Reflist}}

{{DEFAULTSORT:Mean Value Theorem (Divided Differences)}} Category:Finite differences