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

Abstract

If the functions \(f,g:I\rightarrow \mathbb{R}\) are differentiable on the

interval \(I\subseteq \mathbb{R}\), \(a\in I,\) then there exists a function \(\bar{c}:I\rightarrow I\) such that

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

In this paper we study the differentiability of the function \(\bar{c}\), 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
$$ f^{\left( n\right) }\left( a\right) g^{\left( 1\right) }\left( a\right) \neq f^{\left( 1\right) }\left( a\right) g^{\left( n\right) }\left( a\right). $$