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, ..., xn in the domain of an n-times differentiable function f there exists an interior point
ξ ∈ ( min { x 0 , … , x n } , max { x 0 , … , x n } ) {\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:
f [ x 0 , … , x n ] = f ( n ) ( ξ ) n ! . {\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.
Proof
Let P {\displaystyle P} be the Lagrange interpolation polynomial for f at x0, ..., xn. Then it follows from the Newton form of P {\displaystyle P} that the highest order term of P {\displaystyle P} is f [ x 0 , … , x n ] x n {\displaystyle f[x_{0},\dots ,x_{n}]x^{n}} .
Let g {\displaystyle g} be the remainder of the interpolation, defined by g = f − P {\displaystyle g=f-P} . Then g {\displaystyle g} has n + 1 {\displaystyle n+1} zeros: x0, ..., xn. By applying Rolle's theorem first to g {\displaystyle g} , then to g ′ {\displaystyle g'} , and so on until g ( n − 1 ) {\displaystyle g^{(n-1)}} , we find that g ( n ) {\displaystyle g^{(n)}} has a zero ξ {\displaystyle \xi } . This means that
0 = g ( n ) ( ξ ) = f ( n ) ( ξ ) − f [ x 0 , … , x n ] n ! {\displaystyle 0=g^{(n)}(\xi )=f^{(n)}(\xi )-f[x_{0},\dots ,x_{n}]n!} , f [ x 0 , … , x n ] = f ( n ) ( ξ ) n ! . {\displaystyle 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.
References
de Boor, C. (2005). "Divided differences". Surv. Approx. Theory. 1: 46–69. MR 2221566. /wiki/Carl_R._de_Boor ↩