The abscissae (x-coordinates) of the red circles are stationary points; the blue squares are inflection points.

In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below). The value of the function at a critical point is a critical value.^[1]

More specifically, when dealing with functions of a real variable, a critical point, also known as a 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).^[2] 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).^[3][4] Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined).^[5]

This sort of definition extends to differentiable maps between ${\displaystyle \mathbb {R} ^{m}}$ and ${\displaystyle \mathbb {R} ^{n},}$ 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 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 C is a plane curve, defined by an implicit equation f (x,y) = 0, the critical points of the projection onto the x-axis, parallel to the y-axis are the points where the tangent to C are parallel to the y-axis, that is the points where ${\textstyle {\frac {\partial f}{\partial y}}(x,y)=0}$. 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, f (x), is a value x₀ in the domain of f where f is not differentiable or its derivative is 0 (i.e. ${\displaystyle f'(x_{0})=0}$).^[2] A critical value is the image under f of a critical point. These concepts may be visualized through the graph of f: at a critical point, the graph has a horizontal tangent if you can assign one at all.

Notice how, for a differentiable function, critical point is the same as 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 g(x, y) is a differentiable function of two variables, then g(x,y) = 0 is the implicit equation of a curve. A critical point of such a curve, for the projection parallel to the y-axis (the map (x, y) → x), is a point of the curve where ${\displaystyle {\tfrac {\partial g}{\partial y}}(x,y)=0.}$ This means that the tangent of the curve is parallel to the y-axis, and that, at this point, g does not define an implicit function from x to y (see implicit function theorem). If (x₀, y₀) is such a critical point, then x₀ is the corresponding critical value. Such a critical point is also called a bifurcation point, as, generally, when x varies, there are two branches of the curve on a side of x₀ and zero on the other side.

It follows from these definitions that a differentiable function f (x) has a critical point x₀ with critical value y₀, if and only if (x₀, y₀) is a critical point of its graph for the projection parallel to the x-axis, with the same critical value y₀. If f is not differentiable at x₀ due to the tangent becoming parallel to the y-axis, then x₀ is again a critical point of f, but now (x₀, y₀) is a critical point of its graph for the projection parallel to y-axis.

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

Critical points of an implicit curve

Critical points play an important role in the study of plane curves defined by implicit equations, in particular for sketching them and determining their 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 C defined by an implicit equation ${\displaystyle f(x,y)=0}$, where f is a differentiable function 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 ${\displaystyle \pi _{y}}$ and ${\displaystyle \pi _{x}}$, defined by ${\displaystyle \pi _{y}((x,y))=x}$ and ${\displaystyle \pi _{x}((x,y))=y,}$ 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 C is critical for ${\displaystyle \pi _{y}}$, if the tangent to C exists and is parallel to the y-axis. In that case, the images by ${\displaystyle \pi _{y}}$ of the critical point and of the tangent are the same point of the x-axis, called the critical value. Thus a point of C is critical for ${\displaystyle \pi _{y}}$ if its coordinates are a solution of the system of equations:

${\displaystyle f(x,y)={\frac {\partial f}{\partial y}}(x,y)=0}$

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 ${\displaystyle \pi _{x}}$ is similar. If C is the graph of a function ${\displaystyle y=g(x)}$, then (x, y) is critical for ${\displaystyle \pi _{x}}$ if and only if x is a critical point of g, and that the critical values are the same.

Some authors define the critical points of C as the points that are critical for either ${\displaystyle \pi _{x}}$ or ${\displaystyle \pi _{y}}$, although they depend not only on C, 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

${\displaystyle f(x,y)={\frac {\partial f}{\partial x}}(x,y)={\frac {\partial f}{\partial y}}(x,y)=0}$,

and are thus solutions of either system of equations characterizing the critical points. With this more general definition, the critical points for ${\displaystyle \pi _{y}}$ are exactly the points where the implicit function theorem does not apply.

Several variables

For a function of several real variables, a point P (that is a set of values for the input variables, which is viewed as a point in ${\displaystyle \mathbb {R} ^{n}}$) is critical if it is a point where the gradient 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 a local maximum, a local minimum or a 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, 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 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 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. 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.

Critical point of a differentiable map

Given a differentiable map ${\displaystyle f:\mathbb {R} ^{m}\to \mathbb {R} ^{n},}$ the critical points of f are the points of ${\displaystyle \mathbb {R} ^{m},}$ where the rank of the Jacobian matrix of f is not maximal.^[6] 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.

Some authors^[7] give a slightly different definition: a critical point of f is a point of ${\displaystyle \mathbb {R} ^{m}}$ where the rank of the Jacobian matrix of f is less than n. With this convention, all points are critical when m < n.

These definitions extend to differential maps between differentiable manifolds in the following way. Let ${\displaystyle f:V\to W}$ be a differential map between two manifolds V and W of respective dimensions m and n. In the neighborhood of a point p of V and of f (p), charts are diffeomorphisms ${\displaystyle \varphi :V\to \mathbb {R} ^{m}}$ and ${\displaystyle \psi :W\to \mathbb {R} ^{n}.}$ The point p is critical for f if ${\displaystyle \varphi (p)}$ is critical for ${\displaystyle \psi \circ f\circ \varphi ^{-1}.}$ 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 ${\displaystyle \psi \circ f\circ \varphi ^{-1}.}$ If M is a Hilbert manifold (not necessarily finite dimensional) and f is a real-valued function then we say that p is a critical point of f if f is not a submersion at p.^[8]

Application to topology

Critical points are fundamental for studying the topology of manifolds and real algebraic varieties.^[1] 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 ${\displaystyle V}$ be a sub-manifold of ${\displaystyle \mathbb {R} ^{n},}$ and P be a point outside ${\displaystyle V.}$ The square of the distance to P of a point of ${\displaystyle V}$ is a differential map such that each connected component of ${\displaystyle V}$ contains at least a critical point, where the distance is minimal. It follows that the number of connected components of ${\displaystyle V}$ 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