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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, particularly in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
, a stationary point of a
differentiable function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

differentiable function
of one variable is a point on the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

graph
of the function where the function's
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

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 In mathematical analysis, and applications in geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematic ...
, a stationary point is a point on the
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
of the graph where all its
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

partial derivative
s are zero (equivalently, the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

gradient
is zero). 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
is horizontal (i.e.,
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
to the ). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the plane.


Turning points

A turning point is a point at which the derivative changes sign. A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). If the function is differentiable, then a turning point is a stationary point; however not all stationary points are turning points. If the function is twice differentiable, the stationary points that are not turning points are horizontal
inflection point In differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...

inflection point
s. For example, the function x \mapsto x^3 has a stationary point at , which is also an inflection point, but is not a turning point.


Classification

Isolated stationary points of a C^1 real valued function f\colon \mathbb \to \mathbb are classified into four kinds, by the
first derivative testIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
: * a local minimum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive; * a local maximum (maximal turning point or relative maximum) is one where the derivative of the function changes from positive to negative; * a rising
point of inflection . In differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathem ...

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 ; * 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. The first two options are collectively known as "
local extrema In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function (mathematics), function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, ei ...
". 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 extremum—are known as
saddle point In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

saddle point
s. By Fermat's theorem, global extrema must occur (for a C^1 function) on the boundary or at stationary points.


Curve sketching

Determining the position and nature of stationary points aids in
curve sketching In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
of differentiable 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 ''f''''(''x''): * If ''f''''(''x'') < 0, the stationary point at ''x'' is concave down; a maximal extremum. * If ''f''''(''x'') > 0, the stationary point at ''x'' is concave up; 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. 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). A simple example of a point of inflection is the function ''f''(''x'') = ''x''3. There is a clear change of concavity about the point ''x'' = 0, and we can prove this by means of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
. The second derivative of ''f'' is the everywhere-continuous 6''x'', 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\colon \mathbb^ \to \mathbb are those points x0 where the derivative in every direction equals zero, or equivalently, the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

gradient
is zero.


Example

For the function ''f''(''x'') = ''x''4 we have ''f'''(0) = 0 and ''f''''(0) = 0. Even though ''f''''(0) = 0, this point is not a point of inflection. The reason is that the sign of ''f(''x'') changes from negative to positive. For the function ''f''(''x'') = sin(''x'') we have ''f'''(0) ≠ 0 and ''f''''(0) = 0. But this is not a stationary point, rather it is a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of ''f(''x'') does not change; it stays positive. For the function ''f''(''x'') = ''x''3 we have ''f'''(0) = 0 and ''f''''(0) = 0. This is both a stationary point and a point of inflection. This is because the concavity changes from concave downwards to concave upwards and the sign of ''f'''(''x'') does not change; it stays positive.


See also

*
Optimization (mathematics) Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. Optimization problems of sorts arise i ...
* Fermat's theorem *
Derivative testIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
*
Fixed point (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
*
Saddle point In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

Saddle point


References


External links


Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio
at
cut-the-knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electron ...
{{Calculus topics Differential calculus