Movatterモバイル変換

[0]ホーム

Jump to content

Critical point (mathematics)

Edit links

From Wikipedia, the free encyclopedia

Point where the derivative of a function is zero

For other uses, seeCritical point.

This article includes a list ofgeneral references, butit lacks sufficient correspondinginline citations. Please help toimprove this article byintroducing more precise citations.(January 2015) (Learn how and when to remove this message)

The x-coordinates of the red circles arestationary points; the blue squares areinflection points.

Inmathematics, acritical point is theargument of a function where the functionderivative is zero (or undefined, as specified below).The value of the function at a critical point is acritical value.^[1]

More specifically, when dealing withfunctions 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 astationary point) or where the function is notdifferentiable.^[2] Similarly, when dealing withcomplex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is notholomorphic).^[3]^[4] Likewise, for afunction of several real variables, a critical point is a value in its domain where thegradient norm is equal to zero (or undefined).^[5]

This sort of definition extends todifferentiable maps between⁠ $\mathbb {R} ^{m}$ ⁠ and⁠ $\mathbb {R} ^{n},$ ⁠ acritical point being, in this case, a point where therank of theJacobian matrix is not maximal. It extends further to differentiable maps betweendifferentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also calledbifurcation points.In particular, ifC is aplane curve, defined by animplicit equationf (x,y) = 0, the critical points of the projection onto thex-axis, parallel to they-axis are the points where the tangent toC are parallel to they-axis, that is the points where ${\textstyle {\frac {\partial f}{\partial y}}(x,y)=0}$ . In other words, the critical points are those where theimplicit function theorem does not apply.

Critical point of a single variable function

[edit]

Acritical point of a function of a singlereal variable,f (x), is a valuex₀ in thedomain off wheref is notdifferentiable or itsderivative is 0 (i.e. $f'(x_{0})=0$ ).^[2] Acritical value is the image underf of a critical point. These concepts may be visualized through thegraph off: at a critical point, the graph has a horizontaltangent if one can be assigned at all.

Notice how, for adifferentiable function,critical point is the same asstationary 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 acurve (seebelow for a detailed definition). Ifg(x,y) is a differentiablefunction of two variables, theng(x,y) = 0 is theimplicit equation of a curve. Acritical point of such a curve, for the projection parallel to they-axis (the map(x,y) →x), is a point of the curve where ${\tfrac {\partial g}{\partial y}}(x,y)=0.$ This means that the tangent of the curve is parallel to they-axis, and that, at this point,g does not define an implicit function fromx toy (seeimplicit function theorem). If(x₀,y₀) is such a critical point, thenx₀ is the correspondingcritical value. Such a critical point is also called abifurcation point, as, generally, whenx varies, there are two branches of the curve on a side ofx₀ and zero on the other side.

It follows from these definitions that adifferentiable functionf (x) has a critical pointx₀ with critical valuey₀, if and only if(x₀,y₀) is a critical point of its graph for the projection parallel to thex-axis, with the same critical valuey₀. Iff is not differentiable atx₀ due to the tangent becoming parallel to they-axis, thenx₀ is again a critical point off, but now(x₀,y₀) is a critical point of its graph for the projection parallel to they-axis.

For example, the critical points of theunit circle of equation $x^{2}+y^{2}-1=0$ are(0, 1) and(0, -1) for the projection parallel to thex-axis, and(1, 0) and(-1, 0) for the direction parallel to they-axis. If one considers the upper half circle as the graph of the function $f(x)={\sqrt {1-x^{2}}}$ , thenx = 0 is a critical point with critical value 1 due to the derivative being equal to 0, andx = ±1 are critical points with critical value 0 due to the derivative being undefined.

Examples

[edit]

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 numberx₀ for which $2x+2=0.$ This point is aglobal minimum off. The corresponding critical value is $f(-1)=2.$ The graph off is a concave upparabola, 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 they-axis.
The function $f(x)=x^{2/3}$ is defined for allx and differentiable forx ≠ 0, with the derivative $f'(x)={\tfrac {2x^{-1/3}}{3}}$ . Sincef is not differentiable atx = 0 and $f'(x)\neq 0$ otherwise, it is the unique critical point. The graph of the functionf has acusp at this point with vertical tangent. The corresponding critical value is $f(0)=0.$
Theabsolute value function $f(x)=|x|$ is differentiable everywhere except at critical pointx = 0, where it has a global minimum point, with critical value 0.
The function $f(x)={\tfrac {1}{x}}$ has no critical points. The pointx = 0 is not a critical point because it is not included in the function's domain.

Location of critical points

[edit]

By theGauss–Lucas theorem, all of a polynomial function's critical points in thecomplex plane are within theconvex hull of theroots 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 theunit 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

[edit]

Use of the discriminant

[edit]

When the curveC is algebraic, that is when it is defined by a bivariate polynomialf, then thediscriminant is a useful tool to compute the critical points.

Here we consider only the projection $\pi _{y}$ ; Similar results apply to $\pi _{x}$ by exchangingx andy.

Let $\operatorname {Disc} _{y}(f)$ be thediscriminant off viewed as a polynomial iny with coefficients that are polynomials inx. This discriminant is thus a polynomial inx which has the critical values of $\pi _{y}$ among its roots.

More precisely, a simple root of $\operatorname {Disc} _{y}(f)$ 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 thex-coordinate of anasymptote which is parallel to they-axis and is tangent "at infinity" to aninflection 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

[edit]

For afunction of several real variables, a pointP (that is a set of values for the input variables, which is viewed as a point in⁠ $\mathbb {R} ^{n}$ ⁠) iscritical if it is a point where thegradient is zero or undefined.^[5] The critical values are the values of the function at the critical points.

A critical point (where the function is differentiable) may be either alocal maximum, alocal minimum or asaddle point. If the function is at least twice continuously differentiable the different cases may be distinguished by considering theeigenvalues of theHessian matrix of second derivatives.

A critical point at which the Hessian matrix isnonsingular is said to benondegenerate, and the signs of theeigenvalues of the Hessian determine the local behavior of the function. In the case of a function of a single variable, the Hessian is simply thesecond 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 aninflection point, but may also be anundulation point, which may be a local minimum or a local maximum.

For a function ofn variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called theindex of the critical point. A non-degenerate critical point is a local maximum if and only if the index isn, or, equivalently, if the Hessian matrix isnegative definite; it is a local minimum if the index is zero, or, equivalently, if the Hessian matrix ispositive definite. For the other values of the index, a non-degenerate critical point is asaddle point, that is a point which is a maximum in some directions and a minimum in others.

Application to optimization

[edit]

Main article:Mathematical optimization

ByFermat's theorem, all localmaxima 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 asystem of equations, which can be a difficult task. The usualnumerical algorithms are much more efficient for finding local extrema, but cannot certify that all extrema have been found.In particular, inglobal optimization, these methods cannot certify that the output is really the global optimum.

When the function to minimize is amultivariate polynomial, the critical points and the critical values are solutions of asystem 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

[edit]

Given adifferentiable map⁠ $f:\mathbb {R} ^{m}\to \mathbb {R} ^{n},$ ⁠ thecritical points off are the points of⁠ $\mathbb {R} ^{m},$ ⁠ where therank of theJacobian matrix off is not maximal.^[6] The image of a critical point underf is a called a critical value. A point in the complement of the set of critical values is called aregular value.Sard's theorem states that the set of critical values of a smooth map hasmeasure zero.

Some authors^[7] give a slightly different definition: acritical point off is a point of⁠ $\mathbb {R} ^{m}$ ⁠ where the rank of theJacobian matrix off is less thann. With this convention, all points are critical whenm <n.

These definitions extend to differential maps betweendifferentiable manifolds in the following way. Let $f:V\to W$ be a differential map between two manifoldsV andW of respective dimensionsm andn. In the neighborhood of a pointp ofV and off (p),charts arediffeomorphisms $\varphi :V\to \mathbb {R} ^{m}$ and $\psi :W\to \mathbb {R} ^{n}.$ The pointp iscritical forf if $\varphi (p)$ is critical for $\psi \circ f\circ \varphi ^{-1}.$ 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 $\psi \circ f\circ \varphi ^{-1}.$ IfM is aHilbert manifold (not necessarily finite dimensional) andf is a real-valued function then we say thatp is a critical point off iff isnot asubmersion atp.^[8]

Application to topology

[edit]

Critical points are fundamental for studying thetopology ofmanifolds andreal algebraic varieties.^[1] In particular, they are the basic tool forMorse theory andcatastrophe theory.

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

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

References

[edit]

^^a ^bMilnor, John (1963).Morse Theory. Princeton University Press.ISBN 0-691-08008-9.{{cite book}}:ISBN / Date incompatibility (help)
^^a ^bProblems in mathematical analysis. Demidovǐc, Boris P., Baranenkov, G. Moscow(IS): Moskva. 1964.ISBN 0846407612.OCLC 799468131.{{cite book}}:ISBN / Date incompatibility (help)CS1 maint: others (link)
^Stewart, James (2008).Calculus : early transcendentals (6th ed.). Belmont, CA: Thomson Brooks/Cole.ISBN 9780495011668.OCLC 144526840.
^Larson, Ron (2010).Calculus. Edwards, Bruce H., 1946- (9th ed.). Belmont, Calif.: Brooks/Cole, Cengage Learning.ISBN 9780547167022.OCLC 319729593.
^^a ^bAdams, Robert A.; Essex, Christopher (2009).Calculus: A Complete Course.Pearson Prentice Hall. p. 744.ISBN 978-0-321-54928-0.
^Carmo, Manfredo Perdigão do (1976).Differential geometry of curves and surfaces. Upper Saddle River, NJ: Prentice-Hall.ISBN 0-13-212589-7.
^Lafontaine, Jacques (2015).An Introduction to Differential Manifolds. Springer International Publishing.doi:10.1007/978-3-319-20735-3.ISBN 978-3-319-20734-6.
^Serge Lang, Fundamentals of Differential Geometry p. 186,doi:10.1007/978-1-4612-0541-8