This is really a fantastic explanation, but I want to clarify one thing from early on for other readers. If the function isn't changing much locally, it's possible that you are at neither a minimum or a maximum, but rather a saddle point. For instance, every function f(x) = x^k for (positive) odd k has a saddle point at x=0. Both the first and second derivative are 0 at saddle points -- another reason we might be interested in the second derivative and its kin.