View source for Critical point (mathematics)

{{Short description|Point where the derivative of a function is zero}}
[[File:Stationary vs inflection pts.svg|400px|thumb|right|The abscissae (x-coordinates) of the red circles are [[Stationary point|stationary point]]s; the blue squares are [[Inflection point|inflection point]]s.]]

In [[Mathematics|mathematics]], a '''critical point''' is the [[Argument of a function|argument of a function]] where the function [[Derivative|derivative]] is zero (or undefined, as specified below).
The value of the function at a critical point is a '''critical value'''.<ref name="milnor"/>

More specifically, when dealing with [[Function of a real variable|functions of a real variable]], a critical point, also known as a ''[[Stationary point|stationary point]]'', is a point in the domain of the function where the function derivative is equal to zero (or where the function is not [[Differentiable function|differentiable]]).<ref name=":0">{{Cite book|title=Problems in mathematical analysis|date=1964|publisher=Moskva|others=Demidovǐc, Boris P., Baranenkov, G.|isbn=0846407612|location=Moscow(IS)|oclc=799468131}}</ref> Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not not [[Holomorphic function|''holomorphic'']]).<ref>{{Cite book|url=https://archive.org/details/calculusearlytra00stew_1|title=Calculus : early transcendentals|last=Stewart|first=James|date=2008|publisher=Thomson Brooks/Cole|isbn=9780495011668|edition=6th|location=Belmont, CA|oclc=144526840|url-access=registration}}</ref><ref>{{Cite book|title=Calculus|last=Larson|first=Ron|date=2010|publisher=Brooks/Cole, Cengage Learning|others=Edwards, Bruce H., 1946-|isbn=9780547167022|edition= 9th |location=Belmont, Calif.|oclc=319729593}}</ref> Likewise, for a [[Function of several real variables|function of several real variables]], a critical point is a value in its domain where the [[Gradient|gradient]] norm is equal to zero (or undefined).<ref name=":1">{{cite book |last= Adams |first= Robert A. |last2= Essex |first2= Christopher |title= Calculus: A Complete Course |url= https://archive.org/details/calculuscomplete00rada_652 |url-access= limited |publisher= Pearson Prentice Hall |year= 2009 |isbn= 978-0-321-54928-0 |page=[https://archive.org/details/calculuscomplete00rada_652/page/n760 744]}}</ref>

This sort of definition extends to differentiable maps between {{tmath|\R^m}} and {{tmath|\R^n,}} a '''critical point''' being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between [[Differentiable manifold|differentiable manifold]]s, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called ''bifurcation points''.
In particular, if {{mvar|C}} is a [[Plane curve|plane curve]], defined by an implicit equation {{math|1=''f'' (''x'',''y'') = 0}}, the critical points of the projection onto the {{mvar|x}}-axis, parallel to the {{mvar|y}}-axis are the points where the tangent to {{mvar|C}} are parallel to the {{mvar|y}}-axis, that is the points where <math display="inline">\frac{\partial f}{\partial y}(x,y)=0</math>.
In other words, the critical points are those where the [[Implicit function theorem|implicit function theorem]] does not apply.

==Critical point of a single variable function==
A '''critical point''' of a function of a single [[Function of a real variable|real variable]], {{math|''f'' (''x'')}}, is a value {{math|''x''{{sub|0}}}} in the [[Domain of a function|domain]] of {{mvar|f}} where {{mvar|f}} is not [[Differentiable function|differentiable]] or its [[Derivative|derivative]] is 0 (i.e. {{nowrap|<math>f'(x_0) = 0</math>).}}<ref name=":0" /> A '''critical value''' is the image under {{mvar|f}} of a critical point. These concepts may be visualized through the [[Graph of a function|graph]] of {{mvar|f}}: at a critical point, the graph has a horizontal [[Tangent|tangent]] if you can assign one at all.

Notice how, for a [[Differentiable function|differentiable function]], ''critical point'' is the same as [[Stationary point|stationary point]].

Although it is easily visualized on the graph (which is a curve), the notion of critical point of a function must not be confused with the notion of critical point, in some direction, of a curve (see below for a detailed definition). If {{math|''g''(''x'', ''y'')}} is a differentiable [[Function of several real variables|function]] of two variables, then {{math|1=''g''(''x'',''y'') = 0}} is the implicit equation of a curve. A '''critical point''' of such a curve, for the projection parallel to the {{mvar|y}}-axis (the map {{math|(''x'', ''y'') → ''x''}}), is a point of the curve where <math>\tfrac {\partial g}{\partial y}(x,y)=0.</math> This means that the tangent of the curve is parallel to the {{mvar|y}}-axis, and that, at this point, ''g'' does not define an implicit function from {{mvar|x}} to {{mvar|y}} (see [[Implicit function theorem|implicit function theorem]]). If {{math|(''x''{{sub|0}}, ''y''{{sub|0}})}} is such a critical point, then {{math|''x''{{sub|0}}}} is the corresponding '''critical value'''. Such a critical point is also called a '''bifurcation point''', as, generally, when {{mvar|x}} varies, there are two branches of the curve on a side of {{math|''x''{{sub|0}}}} and zero on the other side.

It follows from these definitions that a [[Differentiable function|differentiable function]] {{math|''f'' (''x'')}} has a critical point {{math|''x''{{sub|0}}}} with critical value {{math|''y''{{sub|0}}}}, if and only if {{math|(''x''{{sub|0}}, ''y''{{sub|0}})}} is a critical point of its graph for the projection parallel to the {{mvar|x}}-axis, with the same critical value ''y{{sub|0}}.'' If {{mvar|f}} is not differentiable at {{math|''x''{{sub|0}}}} due to the tangent becoming parallel to the {{mvar|y}}-axis, then {{math|''x''{{sub|0}}}} is again a critical point of {{mvar|f}}, but now {{math|(''x''{{sub|0}}, ''y''{{sub|0}})}} is a critical point of its graph for the projection parallel to {{mvar|y}}-axis.

For example, the critical points of the [[Unit circle|unit circle]] of equation <math>x^2 + y^2 - 1 = 0</math> are (0, 1) and (0, -1) for the projection parallel to the {{mvar|x}}-axis, and (1, 0) and (-1, 0) for the direction parallel to the {{mvar|y}}-axis. If one considers the upper half circle as the graph of the function <math>f(x)= \sqrt{1-x^2},</math> then {{math|1=''x'' = 0}} is a critical point with critical value 1 due to the derivative being equal to 0, and {{math|1=''x'' = ±1}} are critical points with critical value 0 due to the derivative being undefined.

===Examples===
* The function <math>f(x) = x^2 + 2x + 3</math> is differentiable everywhere, with the derivative <math>f'(x)=2x+2.</math> This function has a unique critical point −1, because it is the unique number {{math|''x''{{sub|0}}}} for which <math>2x+2=0.</math> This point is a global minimum of {{mvar|f}}. The corresponding critical value is <math>f(-1) = 2.</math> The graph of {{mvar|f}} is a concave up [[Parabola|parabola]], the critical point is the abscissa of the vertex, where the tangent line is horizontal, and the critical value is the ordinate of the vertex and may be represented by the intersection of this tangent line and the {{mvar|y}}-axis.
* The function <math>f(x) = x^{2/3}</math> is defined for all {{mvar|x}} and differentiable for {{math|''x'' ≠ 0}}, with the derivative <math>f'(x) = \tfrac{2x^{-1/3}}{3}.</math> Since {{mvar|f}} is not differentiable at {{math|1=''x'' = 0}} and <math>f'(x)\neq 0</math> otherwise, it is the unique critical point. The graph of the function {{mvar|f}} has a [[Cusp (singularity)|cusp]] at this point with vertical tangent. The corresponding critical value is <math>f(0)=0.</math>
* The [[Absolute value|absolute value function]] <math>f(x) = |x|</math> is differentiable everywhere except at critical point {{math|1=''x'' = 0}}, where it has a global minimum point, with critical value 0.
* The function <math>f(x) = \tfrac{1}{x}</math> has no critical points. The point {{math|1=''x'' = 0}} is not a critical point because it is not included in the function's domain.

===Location of critical points===
By the [[Gauss–Lucas theorem]], all of a polynomial function's critical points in the [[Complex plane|complex plane]] are within the [[Convex hull|convex hull]] of the [[Zero of a function|roots]] of the function. Thus for a polynomial function with only real roots, all critical points are real and are between the greatest and smallest roots.

[[Sendov's conjecture]] asserts that, if all of a function's roots lie in the [[Unit disk|unit disk]] in the complex plane, then there is at least one critical point within unit distance of any given root.

==Critical points of an implicit curve==
Critical points play an important role in the study of [[Plane curve|plane curve]]s defined by implicit equations, in particular for [[Curve sketching|sketching]] them and determining their [[Topology|topology]]. The notion of critical point that is used in this section, may seem different from that of previous section. In fact it is the specialization to a simple case of the general notion of critical point given below.

Thus, we consider a curve {{math|''C''}} defined by an implicit equation <math>f(x,y)=0</math>, where {{math|''f''}} is a [[Differentiable function|differentiable function]] of two variables, commonly a [[Polynomial#Number of variables|bivariate polynomial]]. The points of the curve are the points of the [[Euclidean plane]] whose Cartesian coordinates satisfy the equation. There are two standard [[Projection (mathematics)|projections]] <math>\pi_y</math> and <math>\pi_x</math>, defined by <math>\pi_y ((x,y))=x</math> and <math>\pi_x ((x,y))=y,</math> that map the curve onto the coordinate axes. They are called the ''projection parallel to the y-axis'' and the ''projection parallel to the x-axis'', respectively.

A point of {{math|''C''}} is '''critical for''' <math>\pi_y</math>, if the [[Tangent|tangent]] to {{math|''C''}} exists and is parallel to the ''y''-axis. In that case, the [[Image (mathematics)|image]]s by <math>\pi_y</math> of the critical point and of the tangent are the same point of the ''x''-axis, called the '''critical value'''. Thus a point of {{math|''C''}} is critical for <math>\pi_y</math> if its coordinates are a solution of the [[System of equations|system of equations]]:

:<math>f(x,y)=\frac{\partial f}{\partial y}(x,y)=0</math>

This implies that this definition is a special case of the general definition of a critical point, which is given below.

The definition of a critical point for <math>\pi_x</math> is similar. If {{math|''C''}} is the [[Graph of a function|graph of a function]] <math>y=g(x)</math>, then {{math|(''x'', ''y'')}} is critical for <math>\pi_x</math> if and only if {{math|''x''}} is a critical point of {{math|''g''}}, and that the critical values are the same.

Some authors define the '''critical points''' of {{math|''C''}} as the points that are critical for either <math>\pi_x</math> or <math>\pi_y</math>, although they depend not only on {{math|''C''}}, but also on the choice of the coordinate axes. It depends also on the authors if the [[Singular point of a curve|singular points]] are considered as critical points. In fact the singular points are the points that satisfy

:<math>f(x,y)=\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0</math>,

and are thus solutions of either system of equations characterizing the critical points. With this more general definition, the critical points for <math>\pi_y</math> are exactly the points where the [[Implicit function theorem|implicit function theorem]] does not apply.

===Use of the discriminant===
When the curve {{math|''C''}} is algebraic, that is when it is defined by a bivariate polynomial {{math|''f''}}, then the [[Discriminant|discriminant]] is a useful tool to compute the critical points.

Here we consider only the projection <math>\pi_y</math>; Similar results apply to <math>\pi_x</math> by exchanging {{math|''x''}} and {{math|''y''}}.

Let
<math>\operatorname{Disc}_y(f)</math>
be the [[Discriminant|discriminant]] of {{math|''f''}} viewed as a polynomial in {{math|''y''}} with coefficients that are polynomials in {{math|''x''}}. This discriminant is thus a polynomial in {{math|''x''}} which has the critical values of <math>\pi_y</math> among its roots.

More precisely, a simple root of <math>\operatorname{Disc}_y(f)</math> is either a critical value of <math>\pi_y</math> such the corresponding critical point is a point which is not singular nor an inflection point, or the {{math|''x''}}-coordinate of an [[Asymptote|asymptote]] which is parallel to the {{math|''y''}}-axis and is tangent "at infinity" to an [[Inflection point|inflection point]] (inflexion asymptote).

A multiple root of the discriminant correspond either to several critical points or inflection asymptotes sharing the same critical value, or to a critical point which is also an inflection point, or to a singular point.

==Several variables==
For a [[Function of several real variables|function of several real variables]], a point {{mvar|P}} (that is a set of values for the input variables, which is viewed as a point in {{nowrap|{{tmath|\R^n}})}} is '''critical''' if it is a point where the [[Gradient|gradient]] is zero or undefined.<ref name=":1" /> The critical values are the values of the function at the critical points.

A critical point (where the function is differentiable) may be either a local maximum, a local minimum or a [[Saddle point|saddle point]]. If the function is at least twice continuously differentiable the different cases may be distinguished by considering the eigenvalues of the [[Hessian matrix]] of second derivatives.

A critical point at which the Hessian matrix is nonsingular is said to be ''nondegenerate'', and the signs of the eigenvalues of the Hessian determine the local behavior of the function. In the case of a function of a single variable, the Hessian is simply the [[Second derivative|second derivative]], viewed as a 1×1-matrix, which is nonsingular if and only if it is not zero. In this case, a non-degenerate critical point is a local maximum or a local minimum, depending on the sign of the second derivative, which is positive for a local minimum and negative for a local maximum. If the second derivative is null, the critical point is generally an [[Inflection point|inflection point]], but may also be an undulation point, which may be a local minimum or a local maximum.

For a function of {{mvar|n}} variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the ''index'' of the critical point. A non-degenerate critical point is a local maximum if and only if the index is {{mvar|n}}, or, equivalently, if the Hessian matrix is negative definite; it is a local minimum if the index is zero, or, equivalently, if the Hessian matrix is [[Positive-definite matrix|positive definite]]. For the other values of the index, a non-degenerate critical point is a [[Saddle point|saddle point]], that is a point which is a maximum in some directions and a minimum in others.

===Application to optimization===
{{Main|Mathematical optimization}}

By [[Fermat's theorem (stationary points)|Fermat's theorem]], all local [[Maxima and minima|maxima and minima]] of a continuous function occur at critical points. Therefore, to find the local maxima and minima of a differentiable function, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros. This requires the solution of a system of  equations, which can be a difficult task. The usual numerical algorithms are much more efficient for finding local extrema, but cannot certify that all extrema have been found.
In particular, in [[Global optimization|global optimization]], these methods cannot certify that the output is really the global optimum.

When the function to minimize is a multivariate polynomial, the critical points and the critical values are solutions of a [[System of polynomial equations|system of polynomial equations]], and modern algorithms for solving such systems provide competitive certified methods for finding the global minimum.

==Critical point of a differentiable map==
Given a differentiable map {{tmath|f: \R^m \to \R^n,}} the '''critical points''' of {{mvar|f}} are the points of {{tmath|\R^m,}} where the [[Rank (linear algebra)|rank]] of the Jacobian matrix of {{mvar|f}} is not maximal.<ref>{{cite book |last=Carmo |first=Manfredo Perdigão do |title=Differential geometry of curves and surfaces |location=Upper Saddle River, NJ |publisher=Prentice-Hall |year=1976 |isbn=0-13-212589-7 }}</ref> The image of a critical point under {{mvar|f}} is a called a [[Critical value|critical value]].  A point in the complement of the set of critical values is called a '''regular value'''. [[Sard's theorem]] states that the set of critical values of a smooth map has measure zero.

Some authors<ref>{{cite book |last=Lafontaine |first=Jacques |title=An Introduction to Differential Manifolds  |publisher=Springer International Publishing|year=2015 |isbn=978-3-319-20734-6|doi=10.1007/978-3-319-20735-3 }}</ref> give a slightly different definition: a '''critical point''' of {{mvar|f}} is a point of {{tmath|\R^m}} where the rank of the Jacobian matrix of {{mvar|f}} is less than {{mvar|n}}. With this convention, all points are critical when {{math|''m'' < ''n''}}.

These definitions extend to differential maps between [[Differentiable manifold|differentiable manifold]]s in the following way. Let <math>f:V \to W</math> be a differential map between two manifolds {{mvar|V}} and {{mvar|W}} of respective dimensions {{mvar|m}} and {{mvar|n}}. In the neighborhood of a point {{mvar|p}} of {{math|''V''}} and of {{math|''f'' (''p'')}}, [[Manifold#Charts|charts]] are [[Diffeomorphism|diffeomorphism]]s <math>\varphi : V \to \R^m</math> and <math>\psi : W \to \R^n.</math> The point {{mvar|p}} is '''critical''' for {{mvar|f}} if <math>\varphi(p)</math> is critical for <math>\psi \circ f \circ \varphi^{-1}.</math> This definition does not depend on the choice of the charts because the transitions maps being diffeomorphisms, their Jacobian matrices are invertible and multiplying by them does not modify the rank of the Jacobian matrix of <math>\psi \circ f \circ \varphi^{-1}.</math> If {{mvar|M}} is a [[Hilbert manifold]] (not necessarily finite dimensional) and {{mvar|f}} is a real-valued function then we say that {{mvar|p}} is a critical point of {{mvar|f}} if {{mvar|f}} is ''not'' a [[Submersion (mathematics)|submersion]] at {{mvar|p}}.<ref>[[Biography:Serge Lang|Serge Lang]], Fundamentals of Differential Geometry p. 186,{{doi|10.1007/978-1-4612-0541-8}}</ref>

==Application to topology==
Critical points are fundamental for studying the [[Topology|topology]] of [[Manifold|manifold]]s and [[Real algebraic geometry|real algebraic varieties]].<ref name="milnor">{{cite book|last=Milnor|first=John|title=Morse Theory|publisher=Princeton University Press|year=1963|isbn=0-691-08008-9}}</ref> In particular, they are the basic tool for [[Morse theory]] and [[Catastrophe theory|catastrophe theory]].

The link between critical points and topology already appears at a lower level of abstraction. For example, let <math>V</math> be a sub-manifold of <math>\mathbb R^n,</math> and {{math|P}} be a point outside <math>V.</math> The square of the distance to {{math|P}} of a point of <math>V</math> is a differential map such that each connected component of <math>V</math> contains at least a critical point, where the distance is minimal. It follows that the number of connected components of <math>V</math> is bounded above by the number of critical points.

In the case of real algebraic varieties, this observation associated with [[Bézout's theorem]] allows us to bound the number of connected components by a function of the degrees of the polynomials that define the variety.

==See also==
* [[Singular point of a curve]]
* [[Singularity theory]]
* [[Gauss–Lucas theorem]]

==References==
{{reflist}}

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[[Category:Multivariable calculus]]
[[Category:Smooth functions]]
[[Category:Singularity theory]]

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