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 :)

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