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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.[1][2][3] 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.
- ^ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (3rd ed.). New York: McGraw-Hill. p. 236. ISBN 0-07-010813-7.
- ^ Cite error: The named reference
Saddlerwas invoked but never defined (see the help page). - ^ "Turning points and stationary points". TCS FREE high school mathematics 'How-to Library'. Retrieved 30 October 2011.