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:

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