Critical locus
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Critical point is a wide term used in many branches of mathematics. When dealing with functions of a real variable, a critical point is a point in the domain of the function where the function is either not
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
or the derivative is equal to zero. When dealing with
complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
s, a critical point is, similarly, a point in the function's domain where it is either not
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
or the derivative is equal to zero. Likewise, for a
function of several real variables In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
, a critical point is a value in its domain where the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is undefined or is equal to zero. The value of the function at a critical point is a critical value. This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called '' bifurcation points''. In particular, if is a
plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
, defined by an
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
, the critical points of the projection onto the -axis, parallel to the -axis are the points where the tangent to are parallel to the -axis, that is the points where \frac(x,y)=0 In other words, the critical points are those where the implicit function theorem does not apply. The notion of a ''critical point'' allows the mathematical description of an
astronomical Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, galaxi ...
phenomenon that was unexplained before the time of
Copernicus Nicolaus Copernicus (; pl, Mikołaj Kopernik; gml, Niklas Koppernigk, german: Nikolaus Kopernikus; 19 February 1473 – 24 May 1543) was a Renaissance polymath, active as a mathematician, astronomer, and Catholic canon, who formulated ...
. A stationary point in the orbit of a planet is a point of the trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop before restarting in the other direction. This occurs because of a critical point of the projection of the orbit into the ecliptic circle.


Critical point of a single variable function

A critical point of a function of a single real variable, , is a value in the domain of where it is not
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
or its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is 0 (i.e. A critical value is the image under of a critical point. These concepts may be visualized through 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 discre ...
of : at a critical point, the graph has a horizontal
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
if you can assign one at all. Notice how, for a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
, ''critical point'' is the same as
stationary point 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. Informally, it is a point where the function "stops" in ...
. 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 In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
(see below for a detailed definition). If is a differentiable
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
of two variables, then is the
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
of a curve. A critical point of such a curve, for the projection parallel to the -axis (the map ), is a point of the curve where \tfrac (x,y)=0. This means that the tangent of the curve is parallel to the -axis, and that, at this point, ''g'' does not define an implicit function from to (see implicit function theorem). If is such a critical point, then is the corresponding critical value. Such a critical point is also called a bifurcation point, as, generally, when varies, there are two branches of the curve on a side of and zero on the other side. It follows from these definitions that a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
has a critical point with critical value , if and only if is a critical point of its graph for the projection parallel to the -axis, with the same critical value ''y.'' If is not differentiable at due to the tangent becoming parallel to the -axis, then is again a critical point of , but now is a critical point of its graph for the projection parallel to -axis. For example, the critical points of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
of equation x^2 + y^2 - 1 = 0 are (0, 1) and (0, -1) for the projection parallel to the -axis, and (1, 0) and (-1, 0) for the direction parallel to the -axis. If one considers the upper half circle as the graph of the function f(x)= \sqrt, then is a critical point with critical value 1 due to the derivative being equal to 0, and are critical points with critical value 0 due to the derivative being undefined.


Examples

* The function f(x) = x^2 + 2x + 3 is differentiable everywhere, with the derivative f'(x)=2x+2. This function has a unique critical point −1, because it is the unique number for which 2x+2=0. This point is a global minimum of . The corresponding critical value is f(-1) = 2. The graph of is a concave up
parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exact ...
, 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 -axis. * The function f(x) = x^ is defined for all and differentiable for , with the derivative f'(x) = \tfrac. Since is not differentiable at and f'(x)\neq 0 otherwise, it is the unique critical point. The graph of the function has a
cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...
at this point with vertical tangent. The corresponding critical value is f(0)=0. * The absolute value function f(x) = , x, is differentiable everywhere except at critical point , where it has a global minimum point, with critical value 0. * The function f(x) = \tfrac has no critical points. The point is not a critical point because it is not included in the function's domain.


Location of critical points

By the
Gauss–Lucas theorem In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial ''P'' and the roots of its derivative ''P′''. The set of roots of a real or complex polynomial is a set of points ...
, all of a polynomial function's critical points in the complex plane are within the convex hull of the
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
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 In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. The ...
asserts that, if all of a function's roots lie in the 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 In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
s defined by
implicit equation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
s, in particular for sketching them and determining their
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. 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 defined by an implicit equation f(x,y)=0, where is a
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
of two variables, commonly a bivariate polynomial. The points of the curve are the points of the Euclidean plane whose Cartesian coordinates satisfy the equation. There are two standard projections \pi_y and \pi_x, defined by \pi_y ((x,y))=x and \pi_x ((x,y))=y, that map the curve onto the
coordinate axes A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
. They are called the ''projection parallel to the y-axis'' and the ''projection parallel to the x-axis'', respectively. A point of is critical for \pi_y, if the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
to exists and is parallel to the ''y''-axis. In that case, the images by \pi_y of the critical point and of the tangent are the same point of the ''x''-axis, called the critical value. Thus a point is critical for \pi_y if its coordinates are solution of the system of equations: :f(x,y)=\frac(x,y)=0 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 \pi_x is similar. If is the graph of a function y=g(x), then is critical for \pi_x if and only if is a critical point of , and that the critical values are the same. Some authors define the critical points of as the points that are critical for either \pi_x or \pi_y, although they depend not only on , but also on the choice of the coordinate axes. It depends also on the authors if the singular points are considered as critical points. In fact the singular points are the points that satisfy :f(x,y)=\frac(x,y)=\frac(x,y)=0, and are thus solutions of either system of equations characterizing the critical points. With this more general definition, the critical points for \pi_y are exactly the points where the implicit function theorem does not apply.


Use of the discriminant

When the curve is algebraic, that is when it is defined by a bivariate polynomial , then the discriminant is a useful tool to compute the critical points. Here we consider only the projection \pi_y; Similar results apply to \pi_x by exchanging and . Let \operatorname_y(f) be the discriminant of viewed as a polynomial in with coefficients that are polynomials in . This discriminant is thus a polynomial in which has the critical values of \pi_y among its roots. More precisely, a simple root of \operatorname_y(f) is either a critical value of \pi_y such the corresponding critical point is a point which is not singular nor an inflection point, or the -coordinate of an
asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
which is parallel to the -axis and is tangent "at infinity" to an
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
(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 In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
, a point (that is a set of values for the input variables, which is viewed as a point in is critical if it is a point where the gradient is undefined or the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is zero. 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 In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
, a
local minimum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
or a saddle point. If the function is at least twice continuously differentiable the different cases may be distinguished by considering the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s of the
Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...
of second derivatives. A critical point at which the Hessian matrix is
nonsingular In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplica ...
is said to be ''nondegenerate'', and the signs of the
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s 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 In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
, 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 In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
, but may also be an undulation point, which may be a local minimum or a local maximum. For a function of 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 , 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. For the other values of the index, a non-degenerate critical point is a saddle point, that is a point which is a maximum in some directions and a minimum in others.


Application to optimization

By Fermat's theorem, all local
maxima and minima In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
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 does not work well in practice because it requires the solution of a nonlinear system of
simultaneous equations In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
, which is 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 is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the max ...
, these methods cannot certify that the output is really the global optimum. When the function to minimize is a
multivariate polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
, the critical points and the critical values are solutions of a
system of polynomial equations A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field . A ''solution'' of a polynomial system is a set of values for the ...
, 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 the critical points of are the points of where the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of the Jacobian matrix of is not maximal. The image of a critical point under is a called a
critical value Critical value may refer to: *In differential topology, a critical value of a differentiable function between differentiable manifolds is the image (value of) ƒ(''x'') in ''N'' of a critical point ''x'' in ''M''. *In statistical hypothesis ...
. 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 In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
. Some authors give a slightly different definition: a critical point of is a point of where the rank of the Jacobian matrix of is less than . With this convention, all points are critical when . These definitions extend to differential maps between differentiable manifolds in the following way. Let f:V \to W be a differential map between two manifolds and of respective dimensions and . In the neighborhood of a point of and of , charts are
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s \varphi : V \to \R^m and \psi : W \to \R^n. The point is critical for if \varphi(p) is critical for \psi \circ f \circ \varphi^. 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 \psi \circ f \circ \varphi^. If is a Hilbert manifold (not necessarily finite dimensional) and is a real-valued function then we say that is a critical point of if is ''not'' a submersion at . Serge Lang, Fundamentals of Differential Geometry p. 186,


Application to topology

Critical points are fundamental for studying the
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
of manifolds and real algebraic varieties. In particular, they are the basic tool for
Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
and catastrophe theory. The link between critical points and topology already appears at a lower level of abstraction. For example, let V be a sub-manifold of \mathbb R^n, and be a point outside V. The square of the distance to of a point of V is a differential map such that each connected component of V contains at least a critical point, where the distance is minimal. It follows that the number of connected components of V is bounded above by the number of critical points. In the case of real algebraic varieties, this observation associated with
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
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 In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic cur ...
*
Singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
*
Gauss–Lucas theorem In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial ''P'' and the roots of its derivative ''P′''. The set of roots of a real or complex polynomial is a set of points ...


References

{{DEFAULTSORT:Critical Point (Mathematics) Multivariable calculus Smooth functions Singularity theory