Applications of the Mean Value Theorem What are some interesting applications of the Mean Value Theorem for derivatives? Both the 'extended' or 'non-extended' versions as seen here are of interest.
So far I've seen some trivial applications like finding the number of roots of a polynomial equation. What are some more interesting applications of it?
I'm asking this as I'm not exactly sure why MVT is so important - so examples which focus on explaining that would be appreciated.

An application which is used a lot in any calculus course:
The derivative of a differentiable function vanishes at an extremum(ie maximum or minimum) point.
