I remember seeing this proof in one of my first-year analysis lectures at university. It was then that I started to appreciate that, in mathematics, it is essential to be able to prove things, even though it might seem blindingly obvious. I remember studying the Euler-Lagrange equations the following year. It was only then that I was able to prove that the shortest distance between two points was a straight line (tongue in cheek comment really - I know this was just to show that the E-L method works in practise).
It should be noted that this method is only guaranteed to find one value for c, provided it exists. The function could equal zero at a number of points in the interval. Other methods would need to be used to find all the values of c where f(c) = 0, if there is more than one.
