Stationary point

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 derivative is zero (equivalently, the gradient is zero): where the function "stops" increasing or decreasing (hence the name).

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 xy plane.

The term is mostly used in one dimension, which this article discusses: stationary points in higher dimensions are usually referred to as critical points; see there for higher dimensional discussion.

Versus critical point

The term "critical point" is often 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.

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.

Note that 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.

Classification

See also: maxima and minima

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

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

• 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.

Curve sketching

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.

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) 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.

See also

• Fermat's theorem
• First derivative test
• Second derivative test
• Higher order derivative test
• Fermat's theorem (stationary points)
• Fixed point (mathematics)

External links

• Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio at cut-the-knot