Mean Value Theorem

The Mean Value Theorem is a generalization of Rolle's Theorem: 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. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that

The Mean Value Theorem claims the existence of a point at which the tangent is parallel to the secant joining (a, f(a)) and (b, f(b)). Rolle's theorem is clearly a particular case of the MVT in which f satisfies an additional condition, f(a) = f(b).)

• Let f be a differentiable function such that the derivative f' is positive on the closed interval [a, b]. Then f is increasing on [a, b].
• Let f be a differentiable function such that the derivative f' is negative on the closed interval [a, b]. Then f is decreasing on [a, b].
  • f (x)=0 everywhere on I if and only if f is constant on I.
  • If f (x)=g (x) for all x on I, then f and g differ at most by a constant on I.
  • If f (x) 0 for all x on I, then f is increasing on I; If f (x) 0 for all x on I, then f is decreasing on
Consequences of the Mean Value Theorem

The Mean Value Theorem is behind many of the important results in calculus. The following statements, in which we assume f is differentiable on an open interval I, are consequences of the Mean Value Theorem: