Stationary Point - Stationary Points, Critical Points and Turning Points

Stationary Points, Critical Points and Turning Points

The term "critical point" may be confused with "stationary point". Critical point is more general: a critical point is either a stationary point or a point where the derivative is not defined. Thus a stationary point is always a critical point, but a critical point is not always a stationary point: it might also be a non-differentiable point. For a smooth function, these are interchangeable, hence the confusion; when a function is understood to be smooth, one can refer to stationary points as critical points, but when it may be non-differentiable, one should distinguish these notions. There is also a different definition of critical point in higher dimensions, where the derivative does not have full rank, but is not necessarily zero; this is not analogous to stationary points, as the function may still be changing in some direction.

A turning point is a point at which the derivative changes sign. A turning point may be either a local maximum or a minimum point. If the function is smooth, then the turning point must be a stationary point, however not all stationary points are turning points, for example has a stationary point at x=0, but the derivative doesn't change sign as there is a point of inflexion at x=0.