Mean value theorem (divided differences)

In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives.

Statement of the theorem

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

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

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

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

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

Proof

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 term of $P$ is $f[x_{0},\dots ,x_{n}](x-x_{n-1})\dots (x-x_{1})(x-x_{0})$.

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

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

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

Applications

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