Second order differentiability of the intermediate-point function in Cauchy's mean-value theorem

Abstract

If the functions $f,g:I\to \mathbb{R}$ are differentiable on the

interval $I\subseteq \mathbb{R}$, $a\in I,$ then there exists a function $\overline{c}:I\to I$ such that

$$[f\left(x\right)-f\left(a\right)]g^{\left(1\right)}\left(\overline{c}\left(x\right)\right)=[g\left(x\right)-g\left(a\right)]f^{\left(1\right)}\left(\overline{c}\left(x\right)\right),\text{ for }x\in I.$$

In this paper we study the differentiability of the function $\overline{c}$, when
$$f^{\left(k\right)}\left(a\right)g^{\left(1\right)}\left(a\right)=f^{\left(1\right)}\left(a\right)g^{\left(k\right)}\left(a\right),\text{ for all }k\in \{1,...,n-1\}$$

and
$$f^{\left(n\right)}\left(a\right)g^{\left(1\right)}\left(a\right)\ne f^{\left(1\right)}\left(a\right)g^{\left(n\right)}\left(a\right).$$