Mean Value Theorem

The Mean Value Theorem is a simplification of Rolle's Theorem:

The Mean Value Theorem 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 the other theorems in Differential and Integral Calculus. It is most frequently derived as the consequence of its own special case -- Rolle's Theorem.

 Let f be a function which is differentiable on closed interval [a, b]. Then there exists a point c in (a, b) so that

f'(c) = f(b) - f(a) / b-a

o    Let f be the differentiable function so that the derivative f ' is positive on the closed interval [a, b]. Then f is increases on [a, b].

o    Let f be a differentiable function so that the derivative f ' is negative on closed interval [a, b]. Then f is decreases on [a, b].

Consequences of the Mean Value Theorem

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

f(x)=0 everywhere on I if and if f is constant on I.

If f(x)=g(x) for all the x on I, then f and g differ at most by the constant on I.

If f(x)0 for all the x on I, then f is increasing on I. 

If f(x)0 for all the x on I, then f is decreasing on I.

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