Looking for the extreme values of a function (maximum and minimum values) can be a daunting task, especially considering that functions are defined at infinitely many (uncountably infinitely many at that) points. Fermat's theorem says that for differentiable functions, the number of possibilities is MUCH smaller—we only need to look at points where the derivative equals zero.
This video introduces the theorem and proves why it is true by considering the definition of the derivative using a limit.
