First Derivative Test - Expii

Let \(f(x)\) be a function. By Fermat's theorem on interior extrema, all local extrema of \(f(x)\) are either stationary points (derivative exists and vanishes) or points of non-differentiability (derivative does not exist). The first derivative test uses the calculus of increasing and decreasing functions to help you distinguish between saddle points and turning points, the two types of stationary points. While turning points correspond to local extrema, saddle points do not.