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Information about Critical Point (mathematics)

Stationary points (red pluses) and inflection points (green circles). It's important to note that the stationary points are critical points, but the inflection points are not.
In mathematics, a critical point (or critical number) is a point on the domain of a function where:
  • one dimension: the derivative (or slope of the line when visualized) is equal to zero or a point where the function ceases to be differentiable.
  • in general: there are two distinct concepts: either the derivative (Jacobian) vanishes, or it is not of full rank (or, in either case, the function in not differentiable); these agree in one dimension.
The value of a function at a critical point is called a critical value ("points" are inputs, "values" are outputs). Elements of the codomain of a function which are not critical values are called regular values.

In One Dimension

There are two situations in which a point becomes a critical point of a function of one variable. The first of which is that the value of the derivative is equal to zero. This point is called a stationary point of the function. An example of this occurring is the function f(x) = x2 + 2x at the value -1, as the function's derivative is f'(x) = 2x + 2, which, when evaluated at -1, equals 0.

The other way a point can be declared a critical point is if the derivative is not defined at that point. An example of this occurring is g(x) = x-1 + x, with its derivative being g'(x)=1 - x-2. It's critical points are 0, as the derivative is not defined there, and both -1 and 1, as the derivative is equal to zero at those points. The function G(x) therefore has 3 critical points, {-1, 0, 1}.

Optimization

See also: maxima and minima


By Fermat's theorem, maxima and minima of a function can occur either at its critical points or at points on its boundary.

A critical point is sometimes not a local maximum or minimum, in which case it is called a saddle point.

Several variables

In this section, functions are assumed to be smooth.

For a smooth function of several real variables, the condition of being a critical point is equivalent to all of its partial derivatives being zero; for a function on a manifold, it is equivalent to the exterior derivative being zero. For a map between spaces of arbitrary finite or infinite dimension, it means that the derivative is zero as a linear map.

If a critical point has a nonsingular Hessian matrix it is called nondegenerate, and the signs of the eigenvalues of the Hessian determine the function's local behavior. 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. In general, the number of negative eigenvalues of a critical point is called its index, and a maximum occurs when all eigenvalues are negative (maximal index: the Hessian is negative definite) and a minimum occurs when all eigenvalues are positive (index zero: the Hessian is positive definite); otherwise it is a saddle point (the Hessian is indefinite (and nonsingular)). Morse theory studies both finite and infinite dimensional manifolds using these ideas.

Gradient vector field

In the presence of a Riemannian metric or a symplectic form, to every smooth function is associated a vector field (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.

Alternative definition (not full rank)

Critical points are also sometimes defined to be points where the derivative of a function is not of maximum rank. Sard's theorem states that the set of critical values, in this sense of critical point, of a differentiable function has measure zero.

See also

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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A spatial point is a concept used to define an exact location in space. It has no volume, area or length, making it a zero dimensional object. Points are used in the basic language of geometry, physics, vector graphics (both 2D and 3D), and many other fields.
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domain is most often defined as the set of values, D for which a function is defined.[1] A function that has a domain N is said to be a function over N, where N is an arbitrary set.
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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derivative is a measurement of how a function changes when the values of its inputs change. Loosely speaking, a derivative can be thought of as how much a quantity is changing at some given point.
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Slope is often used to describe the measurement of the steepness, incline, gradient, or grade of a straight line. A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run
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equal if and only if they are precisely the same in every way. The complementary notion is distinctness. This defines a binary relation, equality, denoted by the sign of equality "=" in such a way that the statement "x = y" means that x
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0 (zero) is both a number and a numerical digit used to represent that number in numerals. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures.
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Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant.

In algebraic geometry the Jacobian of a curve means the Jacobian variety: a group variety associated to the curve, in which the curve can be embedded.
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In differential topology, a critical value of a differentiable function between differentiable manifolds is the image of a critical point.

The basic result on critical values is Sard's lemma.
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In mathematics, the codomain of a function : → is the set .

The domain of is the set .

The range of is the set defined as .
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stationary point is an input to a function where the derivative is zero (equivalently, the gradient is zero): where the function "stops" increasing or decreasing (hence the name).
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maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety (global
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Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat. It gives a method to find local maxima and minima of differentiable functions by showing that every local extremum of the function is a stationary point (the function derivative is zero in that point).
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maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety (global
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In the most general terms, a saddle point for a smooth function (curve, surface or hypersurface) is a point such that the curve/surface/etc. in the neighborhood of this point lies on different sides of the tangent at this point. In certain contexts the definition may vary.
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In mathematics, the exterior derivative operator of differential geometry extends the concept of the differential of a function to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential (coboundary) used to define de
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function. Given the real-valued function



if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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In mathematics, a definite bilinear form is a bilinear form B such that

B(x, x)


has a fixed sign (positive or negative) when x is not 0.
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In mathematics, positive definite may refer to:
  • positive-definite matrix
  • positive-definite function
  • positive definite function on a group
  • positive definite bilinear form

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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
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manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important.
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In Riemannian geometry, a Riemannian manifold (M,g) (with Riemannian metric g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g
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In mathematics, a symplectic manifold is a smooth manifold M equipped with a closed, nondegenerate, 2-form ω called the symplectic form. The study of symplectic manifolds is called symplectic topology.
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vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space.

Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction
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gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
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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
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Sard's lemma, also known as Sard's theorem or the Morse-Sard theorem, is a result of mathematical analysis characterising the image of the critical points of a smooth function F from one Euclidean space to another as having Lebesgue measure 0 (and so,
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