A zero derivative identifies a candidate, and a sign change determines its role.
For f=x³−3ax, a>0 gives critical points ±√a. The derivative changes + to − at −√a and − to + at √a, giving a local maximum and minimum. At a=0, the derivative vanishes at zero without changing sign. For a<0 it never vanishes.
For general functions, critical points also include domain points where a derivative does not exist.