View source for Stationary point

{{Short description|Zero of the derivative of a function}}
{{Distinguish|text=a [[Fixed point (mathematics)|fixed point]] where x ''='' f''(''x'')''}}
{{more footnotes|date=March 2016}}
[[File:Stationary vs inflection pts.svg|350px|thumb|right|The stationary points are the red circles. In this graph, they are all relative maxima or relative minima. The blue squares are [[inflection point]]s.]]

In [[mathematics]], particularly in [[calculus]],  a '''stationary point''' of a [[differentiable function]] of one variable is a point on the [[graph of a function|graph]] of the function where the function's [[derivative]] is zero.<ref>{{cite book|title=Fundamental Methods of Mathematical Economics|url=https://archive.org/details/fundamentalmetho0000chia_h4v2|url-access=registration|last=Chiang|first=Alpha C.|publisher=McGraw-Hill|year=1984|isbn=0-07-010813-7|edition=3rd|location=New York|page=[https://archive.org/details/fundamentalmetho0000chia_h4v2/page/236 236]|author-link=Alpha Chiang}}</ref><ref name=Saddler/><ref name=TCS>{{cite web|title=Turning points and stationary points|url=http://www.teacherschoice.com.au/Maths_Library/Calculus/stationary_points.htm|work=TCS FREE high school mathematics 'How-to Library'|access-date=30 October 2011}}</ref> 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 (mathematics)|surface]] of the graph where all its [[partial derivative]]s are zero (equivalently, the [[gradient]] has zero [[vector norm|norm]]).
The notion of stationary points of a [[real-valued function]] is generalized as ''[[Critical point (mathematics)|critical points]]'' for [[complex-valued function]]s.

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 (geometry)|parallel]] to the [[Abscissa|{{math|''x''}}-axis]]). For a function of two variables, they correspond to the points on the graph where the tangent plane is parallel to the {{math|''xy''}} plane.

The notion of a ''stationary point'' allows the mathematical description of an [[astronomy|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]].

==Turning points==
A '''turning point''' of a [[differentiable function]] is a point at which the derivative has an [[isolated point|isolated zero]] and changes sign at the point.<ref name=Saddler>{{citation|title=Cambridge 2 Unit Mathematics Year 11|first1=David|last1=Saddler|first2=Julia|last2=Shea|first3=Derek|last3=Ward|publisher=Cambridge University Press|year=2011|isbn=9781107679573|contribution=12 B Stationary Points and Turning Points|page=318|contribution-url=https://books.google.com/books?id=wDKLXdzQL5AC&pg=PA318}}</ref> 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 point]]s. For example, the function <math>x \mapsto x^3</math> has a stationary point at {{nowrap|1=''x'' = 0}}, which is also an inflection point, but is not a turning point.<ref name=TCS>{{cite web|title=Turning points and stationary points|url=http://www.teacherschoice.com.au/Maths_Library/Calculus/stationary_points.htm|work=TCS FREE high school mathematics 'How-to Library'|access-date=30 October 2011}}</ref>

==Classification==
[[File:Extrema example original.svg|thumb|250px|A graph in which local extrema and global extrema have been labeled.]]
{{See also|maxima and minima}}

Isolated stationary points of a <math>C^1</math> real valued function <math>f\colon \mathbb{R} \to \mathbb{R}</math> are classified into four kinds, by the [[first derivative test]]:

* 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;
[[Image:Stationary and inflection pts.gif|thumb|left|250px|[[Saddle point]]s (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).]]
* a '''rising [[inflection point|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 [[concave function|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.

The first two options are collectively known as "[[Maxima and minima|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 point]]s.

By [[Fermat's theorem (stationary points)|Fermat's theorem]], global extrema must occur (for a <math>C^1</math> function) on the boundary or at stationary points.

==Curve sketching==
{{Cubic graph special points.svg}}
Determining the position and nature of stationary points aids in [[curve sketching]] of differentiable functions. Solving the equation ''{{prime|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]] ''{{pprime|f}}''(''x''):
* If ''{{pprime|f}}''(''x'') < 0, the stationary point at ''x'' is concave down; a maximal extremum.
* If ''{{pprime|f}}''(''x'') > 0, the stationary point at ''x'' is concave up; a minimal extremum.
* If ''{{pprime|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''<sup>3</sup>. 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 6''x'', and at ''x'' = 0,  ''{{pprime|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 <math>f\colon \mathbb{R}^{n} \to \mathbb{R}</math> are those
points '''x'''<sub>0</sub> where the derivative in every direction equals zero, or equivalently, the [[gradient]] is zero.

===Examples===
For the function ''f''(''x'') = ''x''<sup>4</sup>  we have ''{{prime|f}}''(0) = 0 and ''{{pprime|f}}''(0) = 0. Even though ''{{pprime|f}}''(0) = 0, this point is not a point of inflection.  The reason is that the sign of ''{{prime|f}}''(''x'') changes from negative to positive.

For the function ''f''(''x'') = sin(''x'') we have ''{{prime|f}}''(0) ≠ 0 and  ''{{pprime|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 ''{{prime|f}}''(''x'') does not change; it stays positive.

For the function ''f''(''x'') = ''x''<sup>3</sup> we have ''{{prime|f}}''(0) = 0 and ''{{pprime|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 ''{{prime|f}}''(''x'') does not change; it stays positive.

For the function ''f''(''x'') = 0, one has ''{{prime|f}}''(0) = 0 and ''{{pprime|f}}''(0) = 0. The point 0 is a non-isolated stationary point which is not a turning point nor a horizontal point of inflection as the signs of ''{{prime|f}}''(''x'') and ''{{pprime|f}}''(''x'') do not change.

The function ''f''(''x'') = ''x''<sup>5</sup> sin(1/''x'') for ''x'' ≠ 0, and ''f''(0) = 0, gives an example where ''{{prime|f}}''(''x'') and ''{{pprime|f}}''(''x'') are both continuous, ''{{prime|f}}''(0) = 0 and ''{{pprime|f}}''(0) = 0, and yet ''f''(''x'') does not have a local maximum, a local minimum, nor a point of inflection at 0. So, {{math|0}} is a stationary point that is not isolated.

==See also==
* [[Optimization (mathematics)]]
* [[Fermat's theorem (stationary points)|Fermat's theorem]]
* [[Derivative test]]
* [[Fixed point (mathematics)]]
* [[Saddle point]]

==References==
{{reflist}}

==External links==
* [http://www.cut-the-knot.org/Curriculum/Calculus/FourthDegree.shtml Inflection Points of Fourth Degree Polynomials — a surprising appearance of the golden ratio] at [[cut-the-knot]]

{{Calculus topics}}

[[Category:Differential calculus]]