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 ''x''₀, ..., ''x''_''n'' in the domain of an ''n''-times differentiable function ''f'' there exists an interior point

: $\xi \in (\min\,\max\) \,$

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

: f _0,\dots,x_n= \frac.

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 ''x''₀, ..., ''x''_''n''.
Then it follows from the Newton form of $P$ that the highest order term of $P$ is f _0,\dots,x_n^n.

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

: 0 = g^(\xi) = f^(\xi) - f _0,\dots,x_nn!,
: f _0,\dots,x_n= \frac.

Applications

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

References

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