The Mean Value Theorem states that:

Given f is a function which is continuous and differentiable on the closed interval between a and b. Then there exists a point c in (a, b) such that:

f`(c) = f(b) - f(a)
           b - a

That is... if you have a continuous function f between two points x=a and x=b, there will always be a point x=c on the function where the derivative f`(c) is equal to the gradient of the chord joining f(a) and f(b).

Here is my poor attempt at illustrating this:


 ^           ,
 |         ,'    ,'
 |  f(c) ,'    ,'
 |     .x^`. ,' 
 |   ,'/   .\f(b)   
 | ,' /  ,'  \   /
 |   / ,'     `_/
 |----------------------> x
 |  a  c     b

Oh dear, that is quite poor...

The 'curve' represents our function f(x). The points where f(a) and f(b) are joined by a dotted line, and f(c) is indicated by the x. The tangent to the curve at x=c is also represented by a dotted line. The point of the theorem is that it says there will always be a point x=c in the interval between x=a and x=b where the tangent to the curve at x=c is parallel to that of the line joining the points f(a) and f(b).

This theorem is useful for Convergence Testing and Proving a function has only one root in a given interval, amongst other things. (feel fre to /msg :)

Compare: Rolle's theorem; Intermediate Value Theorem

Props to buo for teaching me this shit so I could node it.