Critical point (mathematics)
Encyclopedia
In calculus
, a critical point of a function
of a real variable is any value in the domain where either the function is not differentiable or its derivative
is 0. The value of the function at a critical point is a critical value of the function. These definitions admit generalizations to functions of several variables, differentiable maps between Rm and Rn, and differentiable maps between differentiable manifold
s.
of a single real variable, ƒ(x), is a value x0 in the domain of ƒ where either the function is not differentiable or its derivative
is 0, ƒ′(x0) = 0. Any value in the codomain
of ƒ that is the image of a critical point under ƒ is a critical value of ƒ. These concepts may be visualized through the graph
of ƒ: at a critical point, either the graph does not admit the tangent
or the tangent is a vertical or horizontal line. In the last case, the derivative is zero and the point is called a stationary point
of the function.
, local maxima and minima
of a function can occur only at its critical points. However, not every stationary point is a maximum or a minimum of the function — it may also correspond to an inflection point
of the graph, as for ƒ(x) = x3 at x = 0, or the graph may oscillate in the neighborhood of the point, as in the case of the function defined by the formulae ƒ(x) = x2sin(1/x) for x ≠ 0 and ƒ(0) = 0, at the point x = 0.
.
For a smooth function of several real variables, the condition of being a critical point is equivalent to all of its partial derivative
s being zero; for a function on a manifold, it is equivalent to its differential
being zero.
If the Hessian matrix
at a critical point is nonsingular then the critical point is called nondegenerate, and the signs of the eigenvalues of the Hessian determine the local behavior of the function. In the case of a real function of a real variable, the Hessian is simply the second derivative
, and nonsingularity is equivalent to being nonzero. A nondegenerate critical point of a single-variable real function is a maximum if the second derivative is negative, and a minimum if it is positive. For a function of n variables, the number of negative eigenvalues of a critical point is called its index, and a maximum occurs when all eigenvalues are negative (index n, the Hessian is negative definite) and a minimum occurs when all eigenvalues are positive (index zero, the Hessian is positive definite); in all other cases, the critical point can be a maximum, a minimum or a saddle point
(index strictly between 0 and n, the Hessian is indefinite). Morse theory
applies these ideas to determination of topology of manifold
s, both of finite and of infinite dimension.
(the gradient
or Hamiltonian vector field
). These vector fields vanish exactly at the critical points of the original function, and thus the critical points are stationary, i.e. constant trajectories of the flow associated to the vector field.
less than n; in particular, every point is critical if m < n. This definition immediately extends to maps between smooth manifolds. The image of a critical point under f 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.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, a critical point of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of a real variable is any value in the domain where either the function is not differentiable or its derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
is 0. The value of the function at a critical point is a critical value of the function. These definitions admit generalizations to functions of several variables, differentiable maps between Rm and Rn, and differentiable maps between differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s.
Definition for single variable functions
A critical point of a functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of a single real variable, ƒ(x), is a value x0 in the domain of ƒ where either the function is not differentiable or its derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
is 0, ƒ′(x0) = 0. Any value in the codomain
Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
of ƒ that is the image of a critical point under ƒ is a critical value of ƒ. These concepts may be visualized through the graph
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
of ƒ: at a critical point, either the graph does not admit the tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
or the tangent is a vertical or horizontal line. In the last case, the derivative is zero and the point is called a stationary point
Stationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
of the function.
Optimization
By Fermat's theoremFermat's theorem (stationary points)
In mathematics, Fermat's theorem is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point...
, local maxima and minima
Maxima and minima
In mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...
of a function can occur only at its critical points. However, not every stationary point is a maximum or a minimum of the function — it may also correspond to an inflection point
Inflection point
In differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...
of the graph, as for ƒ(x) = x3 at x = 0, or the graph may oscillate in the neighborhood of the point, as in the case of the function defined by the formulae ƒ(x) = x2sin(1/x) for x ≠ 0 and ƒ(0) = 0, at the point x = 0.
Examples
- The function ƒ(x) = x2 + 2x + 3 is differentiable everywhere, with the derivative ƒ′(x) = 2x + 2. This function has a unique critical point −1, because it is the unique number x0 for which 2x0 + 2 = 0. This point is a global minimum of ƒ. The corresponding critical value is ƒ(−1) = 2. The graph of ƒ is a concave up parabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, 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 y-axis.
- The function f(x) = x2/3 is defined for all x and differentiable for x ≠ 0, with the derivative ƒ′(x) = 2x−1/3/3. Since ƒ′(x) ≠ 0 for x ≠ 0, the only critical point of ƒ is x = 0. The graph of the function ƒ has a cuspCusp (singularity)In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
at this point with vertical tangent. The corresponding critical value is ƒ(0) = 0.
- The function ƒ(x) = x3 − 3x + 1 is differentiable everywhere, with the derivative ƒ′(x) = 3x2 − 3. It has two critical points, at x = −1 and x = 1. The corresponding critical values are ƒ(−1) = 3, which is a local maximum value, and ƒ(1) = −1, which is a local minimum value of ƒ. This function has no global maximum or minimum. Since ƒ(2) = 3, we see that a critical value may also be attained at a non-critical point. Geometrically, this means that a horizontal tangent line to the graph at one point (x = −1) may intersect the graph at an acute angle at another point (x = 2).
Several variables
In this section, functions are assumed to be smoothSmooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
.
For a smooth function of several real variables, the condition of being a critical point is equivalent to all of its partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
s being zero; for a function on a manifold, it is equivalent to its differential
Differential (calculus)
In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable, then a change in the value of x is often denoted Δx . The differential dx represents such a change, but is infinitely small...
being zero.
If the Hessian matrix
Hessian matrix
In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named...
at a critical point is nonsingular then the critical point is called nondegenerate, and the signs of the eigenvalues of the Hessian determine the local behavior of the function. In the case of a real function of a real variable, the Hessian is simply the second derivative
Second derivative
In calculus, the second derivative of a function ƒ is the derivative of the derivative of ƒ. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is...
, and nonsingularity is equivalent to being nonzero. A nondegenerate critical point of a single-variable real function is a maximum if the second derivative is negative, and a minimum if it is positive. For a function of n variables, the number of negative eigenvalues of a critical point is called its index, and a maximum occurs when all eigenvalues are negative (index n, the Hessian is negative definite) and a minimum occurs when all eigenvalues are positive (index zero, the Hessian is positive definite); in all other cases, the critical point can be a maximum, a minimum or a saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
(index strictly between 0 and n, the Hessian is indefinite). Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
applies these ideas to determination of topology of manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s, both of finite and of infinite dimension.
Gradient vector field
In the presence of a Riemannian metric or a symplectic form, to every smooth function is associated a vector fieldVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
(the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
or Hamiltonian vector field
Hamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations...
). These vector fields vanish exactly at the critical points of the original function, and thus the critical points are stationary, i.e. constant trajectories of the flow associated to the vector field.
Definition for maps
For a differentiable map f between Rm and Rn, critical points are the points where the differential of f is a linear map of rankRank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
less than n; in particular, every point is critical if m < n. This definition immediately extends to maps between smooth manifolds. The image of a critical point under f is a called a critical value
Critical value
-Differential topology:In differential topology, a critical value of a differentiable function between differentiable manifolds is the image ƒ in N of a critical point x in M.The basic result on critical values is Sard's lemma...
. 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.