1.7.4—Stationary points

At a stationary point f′(x)=0. If f″(x)>0 it is a local minimum; if f″(x)<0 a local maximum. When f″=0, inspect the sign of f′ or higher derivatives.

Solve f′=0, calculate coordinates, classify each point and check endpoints if the question asks for an absolute maximum or minimum.

For f=x³−3x, f′=3(x²−1), giving x=±1; f″=6x classifies (−1,2) as a maximum and (1,−2) as a minimum.

Local classification does not compare distant points or endpoints, and f″=0 is inconclusive rather than proof of no turning point.