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 .
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).[1] 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).[2] [3] 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).[4]
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 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, defined by an implicit equation 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.
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 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 if one can be assigned 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 is a differentiable function of two variables, then is the implicit equation 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{\partialg}{\partialy}(x,y)=0.
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 of equation
x2+y2-1=0
f(x)=x2+2x+3
f'(x)=2x+2.
2x+2=0.
f(-1)=2.
f(x)=x2/3
f'(x) ≠ 0
f(0)=0.
f(x)=|x|
f(x)=\tfrac{1}{x}
By the Gauss–Lucas theorem, all of a polynomial function's critical points in the complex plane are within the convex hull of the roots 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 the complex plane, then there is at least one critical point within unit distance of any given root.
See also: Algebraic 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 defined by an implicit equation
f(x,y)=0
\piy
\pix
\piy((x,y))=x
\pix((x,y))=y,
A point of is critical for
\piy
\piy
\piy
| f(x,y)= | \partialf |
| \partialy |
(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
\pix
y=g(x)
\pix
Some authors define the critical points of as the points that are critical for either
\pix
\piy
and are thus solutions of either system of equations characterizing the critical points. With this more general definition, the critical points for
\piy
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
\piy
\pix
Let
\operatorname{Disc}y(f)
\piy
More precisely, a simple root of
\operatorname{Disc}y(f)
\piy
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.
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 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. 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 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.
See main article: Mathematical 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.
Given a differentiable map the critical points of are the points of where the rank of the Jacobian matrix of is not maximal.[5] 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.
Some authors[6] 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\toW
\varphi:V\to\Rm
\psi:W\to\Rn.
\varphi(p)
\psi\circf\circ\varphi-1.
\psi\circf\circ\varphi-1.
Critical points are fundamental for studying the topology of manifolds and real algebraic varieties.[8] 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
Rn,
V.
V
V
V
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.