Stationary point

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Stationary points (red pluses) and inflection points (green circles). The stationary points in this graph are all relative maxima or relative minima.

In mathematics, particularly in calculus, a stationary point is an input to a function where the gradient is zero. For the graph of a one-dimensional function, this corresponds to a point on the graph where the tangent is parallel to the x-axis. For the graph of a two-dimensional function, this corresponds to a point on the graph where the tangent plane is parallel to the x-y plane.

Saddle points (coincident stationary points and inflection points). Here one is rising and one is a falling inflection point.

An inflection point is a point where the concavity changes. A point of inflection is not necessarily a stationary point. All inflection points have the property of f''(x) = 0 but the reverse is not necessarily true.

Stationary points of a real valued function f: R → R are classified into four kinds:

See also: maxima and minima

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.

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 straightforward 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 the sign changes about this point. So x = 0 is a point of inflection.

More generally, the stationary points of a real valued function f: Rⁿ → R are those points x₀ where the derivative in every direction equals zero, or equivalently, the gradient is zero.

[edit] Example

At x₁ we have f' (x) = 0 and f''(x) = 0. Even though f''(x) = 0, this point is not a point of inflexion. The reason is that the sign of f' (x) changes from negative to positive.

At x₂, we have f' (x) $\ne$ 0 and f''(x) = 0. But, x₂ is not a stationary point, rather it is a point of inflexion. This because the concavity changes from concave upwards to concave downwards and the sign of f' (x) does not change; it stays positive.

At x₃ we have f' (x) = 0 and f''(x) = 0. Here, x₃ is both a stationary point and a point of inflexion. This is because the concavity changes from concave upwards to concave downwards and the sign of f' (x) does not change; it stays positive.