# Mean value theorem (divided differences)

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In [mathematical analysis](/source/mathematical_analysis), the '''mean value theorem for divided differences''' generalizes the [mean value theorem](/source/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&ndash;69|authorlink=Carl R. de Boor|mr=2221566}}</ref>

== Statement of the theorem ==

For any ''n''&nbsp;+&nbsp;1 pairwise distinct points ''x''<sub>0</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub> 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](/source/divided_difference) at these points:

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

For ''n''&nbsp;=&nbsp;1, that is two function points, one obtains the simple [mean value theorem](/source/mean_value_theorem).

== Proof ==

Let <math>P</math> be the [Lagrange interpolation polynomial](/source/Lagrange_interpolation_polynomial) for ''f'' at ''x''<sub>0</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>.
Then it follows from the [Newton form](/source/Newton_polynomial) 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''<sub>0</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>.
By applying [Rolle's theorem](/source/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](/source/Stolarsky_mean) to more than two variables.

== References ==
{{Reflist}}

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

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