Rolle's and The Mean Value Theorems
The Mean Value Theorem (MVT, for short) is one of the most frequent subjects in mathematics education literature. It is one of important tools in the mathematician's arsenal, used to prove a host of other theorems in Differential and Integral Calculus. As a curiosity, it is most frequently derived as a consequence of its own special case -- Rolle's theorem. The latter is named after Michel Rolle (1652-1719), a French mathematician who established the now common symbol for the nth root and insisted that -a > -b, for positive a and b, a < b. The feat went against Descartes' teaching and laid the groundwork for the introduction of the ubiquitous number line. Let f be continuous on a closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there is at least one point c in (a, b) where f '(c) = 0. (The tangent to a graph of f where the derivative vanishes is parallel to x-axis, and so is the line joining the two "end" points (a, f(a)) and (b, f(b)) on the