Properties of the intermediate point from the Taylor's theorem

Abstract

If I ⊆ R is an interval, a ∈ I and f : I → R is n 1 times differentiable on I , then, in view of Taylor’s theorem, there exists a function c : I → I such that, for each x ∈ I, f (x) = n−1 ∑ k=0 f (k) (a) k! (x−a) + f (n) (c(x)) n! (x−a) . In this paper we study the behaviour of the derivatives c(p) and θ (p) of the functions c and θ , respectively, when x approaches a, where θ : I →]0,1[ is defined by θ (x) = (c(x)−a)/(x−a) , if x ∈ I\{a} and θ (a) = 1/(n+1) . Mathematics subject classification (2000): 26A24.