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

${\displaystyle \xi \in (\min\{x_{0},\dots ,x_{n}\},\max\{x_{0},\dots ,x_{n}\})\,}$

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

${\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}.}$

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

Let ${\displaystyle P}$ be the Lagrange interpolation polynomial for f at x₀, ..., x_n. Then it follows from the Newton form of ${\displaystyle P}$ that the highest term of ${\displaystyle P}$ is ${\displaystyle f[x_{0},\dots ,x_{n}](x-x_{n-1})\dots (x-x_{1})(x-x_{0})}$.

Let ${\displaystyle g}$ be the remainder of the interpolation, defined by ${\displaystyle g=f-P}$. Then ${\displaystyle g}$ has ${\displaystyle n+1}$ zeros: x₀, ..., x_n. By applying Rolle's theorem first to ${\displaystyle g}$, then to ${\displaystyle g'}$, and so on until ${\displaystyle g^{(n-1)}}$, we find that ${\displaystyle g^{(n)}}$ has a zero ${\displaystyle \xi }$. This means that

${\displaystyle 0=g^{(n)}(\xi )=f^{(n)}(\xi )-f[x_{0},\dots ,x_{n}]n!}$,

${\displaystyle f[x_{0},\dots ,x_{n}]={\frac {f^{(n)}(\xi )}{n!}}.}$

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