Stationary points and the first derivative test

A stationary point has y =0. A sign change from positive to negative gives a maximum; negative to positive gives a minimum. The sign change describes a curve rising then falling, or falling then rising. Merely finding zero slope does not decide its nature. Method Solve the zero-derivative equation in the domain, find each output coordinate, and record the derivative sign just before and after each candidate. Example y=(x-1)4, y =4(x-1)3 changes from negative to positive at x=1, so (1,0) is a minimum. Exam check Give both coordinates and both derivative signs. The second derivative being zero is inconclusive, not proof of no minimum.