On the intermediate point in Cauchy's mean-value theorem

Abstract

If the functions f , g : I → R are differentiable on the interval I ⊆ R , then for each x, a ∈ I there exists a real number θ ∈]0, 1[ such that (f (x) − f (a)) g (1) (a + θ(x − a)) = (g (x) − g (a)) f (1) (a + θ(x − a)). In this paper we study the behaviour of the number θ ∈]0, 1[, when x approaches a .

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

“…In [2], one proves the following theorem Theorem 4. Let I be an interval in R, let a be a point of I and let f, g : I → R be functions which satisfy the following conditions:…”

“…If the function f (1) /g (1) is not injective on I, then for some x ∈ I\{a} there exist several points c x , from the interval with the extremities x and a, such that (1) is true. If for each x ∈ I\{a} we choose one c x from the interval with the extremities x and a which satisfies (1) , then we can also define the function c : I\{a} → I\{a} by formula (2). This function c satisfies (3), too (see [2]).…”

“…If for each x ∈ I\{a} we choose one c x from the interval with the extremities x and a which satisfies (1) , then we can also define the function c : I\{a} → I\{a} by formula (2). This function c satisfies (3), too (see [2]).…”

“…The purpose of this paper is to establish under which circumstances the function c is twice differentiable at the point x = a and to compute its derivatives c (1) (a) and c (2) (a) . Do the derivatives c (1) (a) and c (2) (a) depend upon the functions f and g?…”

“…The purpose of this paper is to establish under which circumstances the function c is twice differentiable at the point x = a and to compute its derivatives c (1) (a) and c (2) (a) . Do the derivatives c (1) (a) and c (2) (a) depend upon the functions f and g? If there exist several functions c which satisfy (3) , do the derivatives of the function c at x = a depend upon the function c we choose?…”