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**Critical point** is a wide term used in many branches of mathematics.

- Critical point of a single variable function
- Examples
- Location of critical points
- Critical points of an implicit curve
- Use of the discriminant
- Several variables
- Application to optimization
- Critical point of a differentiable map
- Application to topology
- See also
- References

When dealing with functions of a real variable, a **critical point** is a point in the domain of the function where the function is either not differentiable or the derivative is equal to zero.^{ [1] } When dealing with complex variables, a **critical point** is, similarly, a point in the function's domain where it is either not holomorphic or the derivative is equal to zero.^{ [2] }^{ [3] } Likewise, for a function of several real variables, a **critical point** is a value in its domain where the gradient is undefined or is equal to zero.^{ [4] }

The value of the function at a critical point is a **critical value**.

This sort of definition extends to differentiable maps between **R**^{m} and **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 . In other words, the critical points are those where the implicit function theorem does not apply.

The notion of a *critical point* allows the mathematical description of an astronomical phenomenon that was unexplained before the time of Copernicus. A **stationary point** in the orbit of a planet is a point of the trajectory of the planet on the celestial sphere, where the motion of the planet seems to stop before restarting in the other direction. This occurs because of a critical point of the projection of the orbit into the ecliptic circle.

A **critical point** of a function of a single real variable, *f*(*x*), is a value *x*_{0} in the domain of *f* where it is not differentiable or its derivative is 0 (*f* ′(*x*_{0}) = 0).^{ [1] } 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 . 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 (*x*_{0}, *y*_{0}) is such a critical point, then *x*_{0} 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 *x*_{0} and zero on the other side.

It follows from these definitions that a differentiable function *f*(*x*) has a critical point *x*_{0} with critical value *y*_{0}, if and only if (*x*_{0}, *y*_{0}) is a critical point of its graph for the projection parallel to the *x*-axis, with the same critical value *y _{0}.* If

For example, the critical points of the unit circle of equation *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 , then *x* = 0 is a critical point with critical value 1 due to the derivative being equal to 0, and x=-1 and x=1 are critical points with critical value 0 due to the derivative being undefined.

- The function
*f*(*x*) =*x*^{2}+ 2*x*+ 3 is differentiable everywhere, with the derivative*f*′(*x*) = 2*x*+ 2. This function has a unique critical point −1, because it is the unique number*x*_{0}for which 2*x*_{0}+ 2 = 0. This point is a global minimum of*f*. The corresponding critical value is*f*(−1) = 2. The graph of*f*is a concave up parabola, 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*) =*x*^{2/3}is defined for all*x*and differentiable for*x*≠ 0, with the derivative*f*′(*x*) = 2*x*^{−1/3}/3. Since*f*is not differentiable at x=0 and*f'(x)≠0*otherwise, it is the unique critical point. The graph of the function*f*has a cusp 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 x=0, where it has a global minimum point, with critical value 0.
- The function
*f*(*x*) = 1/*x*has no critical points. The point x = 0 is not a critical point because it is not included in the function's domain.

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.

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 , 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 and , defined by and 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**, if the tangent to *C* exists and is parallel to the *y*-axis. In that case, the images by of the critical point and of the tangent are the same point of the *x*-axis, called the **critical value**. Thus a point is critical for if its coordinates are solution of the system of equations:

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 is similar. If *C* is the graph of a function , then (*x*, *y*) is critical for 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 or , 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

- ,

and are thus solutions of either system of equations characterizing the critical points. With this more general definition, the critical points for are exactly the points where the implicit function theorem does not apply.

When the curve *C* is algebraic, that is when it is defined by a bivariate polynomial *f*, then the discriminant is a useful tool to compute the critical points.

Here we consider only the projection ; Similar results apply to by exchanging *x* and *y*.

Let be the discriminant of *f* viewed as a polynomial in *y* with coefficients that are polynomials in *x*. This discriminant is thus a polynomial in *x* which has the critical values of among its roots.

More precisely, a simple root of is either a critical value of such the corresponding critical point is a point which is not singular nor an inflection point, or the *x*-coordinate of an asymptote which is parallel to the *y*-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.

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 **R**^{n}) is **critical** if it is a point where the gradient is undefined or the gradient is zero.^{ [4] } 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.

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 does not work well in practice because it requires the solution of a nonlinear system of simultaneous equations, which is 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 *f* from **R**^{m} into **R**^{n}, the **critical points** of *f* are the points of **R**^{m}, where the rank of the Jacobian matrix of *f* is not maximal.^{ [5] } 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^{ [6] } give a slightly different definition: a **critical point** of *f* is a point of **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 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 and The point *p* is **critical** for *f* if is critical for 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 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*.^{ [7] }

Critical points are fundamental for studying the topology of manifolds 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 be a sub-manifold of and P be a point outside The square of the distance to P of a point of is a differential map such that each connected component of contains at least a critical point, where the distance is minimal. It follows that the number of connected components of 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.

In analytic geometry, an **asymptote** of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the *x* or *y* coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.

In mathematics, the **derivative** of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

In mathematics, a **parabola** is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

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. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve *y* = *f*(*x*) at a point *x* = *c* if the line passes through the point (*c*, *f* ) on the curve and has slope *f*'(*c*), where *f*' is the derivative of *f*. A similar definition applies to space curves and curves in *n*-dimensional Euclidean space.

In mathematics, **differential calculus** is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

In geometry, a **normal** is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the **normal line** to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one or its length may represent the curvature of the object ; its algebraic sign may indicate sides.

In vector calculus, the **Jacobian matrix** of a vector-valued function in several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the **Jacobian determinant**. Both the matrix and the determinant are often referred to simply as the **Jacobian** in literature.

In mathematics, a **cubic function** is a function of the form

In mathematics, an **affine algebraic plane curve** is the zero set of a polynomial in two variables. A **projective algebraic plane curve** is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation *h*(*x*, *y*, *t*) = 0 can be restricted to the affine algebraic plane curve of equation *h*(*x*, *y*, 1) = 0. These two operations are each inverse to the other; therefore, the phrase **algebraic plane curve** is often used without specifying explicitly whether it is the affine or the projective case that is considered.

In mathematics, specifically differential calculus, the **inverse function theorem** gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its *derivative is continuous and non-zero at the point*. The theorem also gives a formula for the derivative of the inverse function. In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.

In differential calculus and differential geometry, an **inflection point**, **point of inflection**, **flex**, or **inflection** is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave to convex, or vice versa.

In mathematics, the **Hessian matrix** or **Hessian** is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. 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 after him. Hesse originally used the term "functional determinants".

In mathematics, a **saddle point** or **minimax point** is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero, but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function has a critical point at that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the -direction.

In mathematics, particularly in calculus, a **stationary point** of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, it is a point where the function "stops" increasing or decreasing.

In the mathematical field of algebraic geometry, a **singular point of an algebraic variety***V* is a point *P* that is 'special', in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be **regular**. An algebraic variety which has no singular point is said to be **non-singular** or **smooth**.

In mathematics, a **linear approximation** is an approximation of a general function using a linear function. They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.

In geometry and science, a **cross section** is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space showing points on the surface of the mountains of equal elevation.

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.

In mathematics, a **surface** is a generalization of a plane, which is not necessarily flat – that is, the curvature is not necessarily zero. This is analogous to a curve generalizing a straight line. There are many more precise definitions, depending on the context and the mathematical tools that are used to analyze the surface.

In mathematics, a **cusp**, sometimes called **spinode** in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve.

- 1 2
*Problems in mathematical analysis*. Demidovǐc, Boris P., Baranenkov, G. Moscow(IS): Moskva. 1964. ISBN 0846407612. OCLC 799468131.CS1 maint: others (link) - ↑ 1941-, Stewart, James (2008).
*Calculus : early transcendentals*(6th ed.). Belmont, CA: Thomson Brooks/Cole. ISBN 9780495011668. OCLC 144526840.CS1 maint: numeric names: authors list (link) - ↑ 1941-, Larson, Ron (2010).
*Calculus*. Edwards, Bruce H., 1946- (9th ed.). Belmont, Calif.: Brooks/Cole, Cengage Learning. ISBN 9780547167022. OCLC 319729593.CS1 maint: numeric names: authors list (link) - 1 2 Adams, 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

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