mean value theorem (divided differences)

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

: $\backslash xi\; \backslash in\; (\backslash min\backslash \{x\_0,\backslash dots,x\_n\backslash \},\backslash max\backslash \{x\_0,\backslash dots,x\_n\backslash \})\; \backslash ,$

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

: $f[x\_0,\backslash dots,x\_n]\; =\; \backslash frac\{f^\{(n)\}(\backslash xi)\}\{n!\}.$

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

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 $f[x\_0,\backslash dots,x\_n]x^n$.

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 $g\text{'}$, and so on until $g^\{(n-1)\}$, we find that $g^\{(n)\}$ has a zero $\backslash xi$. This means that

: $0\; =\; g^\{(n)\}(\backslash xi)\; =\; f^\{(n)\}(\backslash xi)\; -\; f[x\_0,\backslash dots,x\_n]\; n!$,

: $f[x\_0,\backslash dots,x\_n]\; =\; \backslash frac\{f^\{(n)\}(\backslash xi)\}\{n!\}.$

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

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Category:Finite differences