The geometric Mean Value Theorem

ABSTRACT

In a previous article published in the American Mathematical Monthly, Tucker (Amer Math Monthly. 1997; 104(3): 231–240) made severe criticism on the Mean Value Theorem and, unfortunately, the majority of calculus textbooks also do not help to improve its reputation. The standard argument for proving it seems to be applying Rolle's theorem to a function like $$\begin{array}{c}\hfill h\left(x\right):=f\left(x\right)-f\left(a\right)-\frac{f\left(b\right)-f\left(a\right)}{b-a}(x-a).\end{array}$$Although short and effective, such reasoning is not intuitive. Perhaps for this reason, Tucker classified the Mean Value Theorem as a technical existence theorem used to prove intuitively obvious statements. Moreover, he argued that there is nothing obvious about the Mean Value Theorem without the continuity of the derivative. Under so unfair discrimination, we felt the need to come to the defense of this beautiful theorem in order to clear up these misunderstandings.

KEYWORDS:

• Mean Value Theorem
• differential calculus
• tangent line

Disclosure statement

No potential conflict of interest was reported by the author.