Stationary point

In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing (hence the name).

For a differentiable function of several real variables, a stationary point is a point on the surface of the graph where all its partial derivatives are zero (equivalently, the gradient has zero norm). The notion of stationary points of a real-valued function is generalized as critical points for complex-valued functions.

Stationary points are easy to visualize on the graph of a function of one variable: they correspond to the points on the graph where the tangent is horizontal (i.e., parallel to the x-axis). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the xy plane.

The notion of a stationary point allows the mathematical description of an astronomical phenomenon that was unexplained before the time of Copernicus. A stationary point is the point in the apparent trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop, before restarting in the other direction (see apparent retrograde motion). This occurs because of the projection of the planet orbit into the ecliptic circle.

A turning point of a differentiable function is a point at which the derivative has an isolated zero and changes sign at the point. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). A turning point is thus a stationary point, but not all stationary points are turning points. If the function is twice differentiable, the isolated stationary points that are not turning points are horizontal inflection points. For example, the function ${\displaystyle x\mapsto x^{3}}$ has a stationary point at x = 0, which is also an inflection point, but is not a turning point.

Isolated stationary points of a ${\displaystyle C^{1}}$ real valued function ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ are classified into four kinds, by the first derivative test:

The first two options are collectively known as "local extrema". Similarly a point that is either a global (or absolute) maximum or a global (or absolute) minimum is called a global (or absolute) extremum. The last two options—stationary points that are not local extrema—are known as saddle points.

By Fermat's theorem, global extrema must occur (for a ${\displaystyle C^{1}}$ function) on the boundary or at stationary points.