HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a critical point is the
argument of a function In mathematics, an argument of a function is a value provided to obtain the function's result. It is also called an independent variable. For example, the binary function f(x,y) = x^2 + y^2 has two arguments, x and y, in an ordered pair (x, y) ...
where the function
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
is zero (or undefined, as specified below). The value of the function at a critical point is a . More specifically, when dealing with functions of a real variable, a critical point is a point in the domain of the function where the function derivative is equal to zero (also known as a ''
stationary point 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. Informally, it is a point where the ...
'') or where the function is not differentiable. 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 ''holomorphic''). Likewise, for a function of several real variables, a critical point is a value in its domain where the
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
norm is equal to zero (or undefined). This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
is not maximal. It extends further to differentiable maps between
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
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 is a plane curve, 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 parallel to the are the points where the tangent to are parallel to the that is the points where In other words, the critical points are those where the implicit function theorem does not apply.


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 is not differentiable or its
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
is 0 (i.e. A critical value is the image under of a critical point. These concepts may be visualized through the graph 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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
if one can be assigned at all. Notice how, for a differentiable function, ''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 a function, graph of the function where the function's derivative is zero. Informally, it is a point where the ...
. 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 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 has a critical point with critical value if and only if is a critical point of its graph for the projection parallel to the with the same critical value 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 the 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 and for the projection parallel to the and and for the direction parallel to the If one considers the upper half circle as the graph of the function 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 exactl ...
, 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 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 bifu ...
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, all of a polynomial function's critical points in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
are within the
convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
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 focusin ...
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 In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
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 curves 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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
. 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 of two variables, commonly a bivariate polynomial. The points of the curve are the points of the
Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
whose
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
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. 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, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to exists and is parallel to the ''y''-axis. In that case, the
image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
s 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 of is critical for \pi_y if its coordinates are a 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 In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane (geometry), plane and often form a P ...
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 : 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 which is parallel to the -axis and is tangent "at infinity" to an 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, 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 In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
is zero or undefined. 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 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), ...
. 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 is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s 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
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
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 . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the secon ...
, 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, 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 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), ...
, 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 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, 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, 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 of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
of is not maximal. The image of a critical point under is a called a 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 In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...
. Some authors give a slightly different definition: a critical point of is a point of where the rank of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
of is less than . With this convention, all points are critical when . These definitions extend to differential maps between
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ...
s 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 differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...
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 Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
of
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s and real algebraic varieties. In particular, they are the basic tool for Morse theory 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 In algebraic geometry, Bézout's theorem is a statement 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 de ...
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 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 ...


References

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