Stationary point

In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number).

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

Notice: Global (or absolute) maxima and minima are sometimes called global (or absolute) maximal (resp. minimal) extrema. While they may occur at stationary points, they are not actually an example of a stationary point. See absolute extremum for more information about this.

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 more straight-forward way of determining the nature of a stationary point is by examining the function values between the stationary points. However, this is limited again in that it works only for functions that are continuous in at least a small interval surrounding the stationary point.

A simple example of a point of inflection is the function f(x) = x³. 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 6x, and at x = 0, f ′ ′ = 0, and that the sign changes about this point. So x = 0 is a point of inflection.