Stationary points are classified into four kinds:

- 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

Determining the position and nature of stationary points aids in curve sketching, especially for continuous functions. Solving the equation *f'*(*x*) = 0 returns the *x*-coordinates of all stationary points; the *y*-coordinates are trivially the function values at those *x*-coordinates.

The specific nature of a stationary point at *x* can in some cases be determined by examining the second derivative *f''*(*x*):

- If
*f''*(*x*) < 0, the stationary point at*x*is a maximal extremum. - If
*f''*(*x*) > 0, the stationary point at*x*is a minimal extremum. - If
*f''*(*x*) = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point provided the function values exist around that point.

A simple example of a point of inflection is the function *f*(*x*) = *x*^{3}. There is a clear change of concavity about the point *x* = 0, and we can prove this by means of calculus. The second derivative of *f* is the everywhere-continuous 6*x*, and at *x* = 0, *f* ′ ′ = 0, and that the sign changes about this point. So *x* = 0 is a point of inflection.