Stationary point

[1][2][3] Informally, it is a point where the function "stops" increasing or decreasing (hence the name).

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).

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.

are classified into four kinds, by the first derivative test: The first two options are collectively known as "local extrema".

Determining the position and nature of stationary points aids in curve sketching of differentiable functions.

The specific nature of a stationary point at x can in some cases be determined by examining the second derivative f″(x): A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points (if the function is defined and continuous between them).

There is a clear change of concavity about the point x = 0, and we can prove this by means of calculus.

More generally, the stationary points of a real valued function

are those points x0 where the derivative in every direction equals zero, or equivalently, the gradient is zero.

The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are inflection points .
A graph in which local extrema and global extrema have been labeled.
Saddle points (stationary points that are neither local maxima nor minima: they are inflection points . The left is a "rising point of inflection" (derivative is positive on both sides of the red point); the right is a "falling point of inflection" (derivative is negative on both sides of the red point).
The roots , stationary points , inflection point and concavity of a cubic polynomial x 3 − 6 x 2 + 9 x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives .