Stationary Point - Classification

Classification

Isolated stationary points of a real valued function are classified into four kinds, by the first derivative test:

• a minimal extremum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive;
• a maximal extremum (maximal turning point or relative maximum) is one where the derivative of the function changes from positive to negative;
• a rising point of inflection (or inflexion) is one where the derivative of the function is positive on both sides of the stationary point; such a point marks a change in concavity
• a falling point of inflection (or inflexion) is one where the derivative of the function is negative on both sides of the stationary point; such a point marks a change in concavity

Notice: Global (or absolute) maxima and minima are sometimes called global (or absolute) maximal (resp. minimal) extrema. By Fermat's theorem, they must occur on the boundary or at critical points, but they do not necessarily occur at stationary points.