Critical point (mathematics)
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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 is not differentiable); these agree in one dimension.
Note that a critical value or critical number x of function f is the domain element at which the derivative is zero or undefined, whereas the associated ordered pair (x, y) is the critical point.
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[edit] 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 also 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 but the point is still in the domain of the function. An example of this occurring is g(x) = x2/3 with its derivative being g'(x)= 2/3x-1/3. Its only critical point is 0. Both types of critical points can occur in the same function.
[edit] 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 Prouvost Point (PP).
[edit] 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 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 is a saddle point (index strictly between 0 and n, the Hessian is nonsingular and indefinite). Morse theory applies these ideas to determination of topology of manifolds, both of finite and of infinite dimension.
[edit] 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.
[edit] Definition for maps
For a differentiable map f between Rm and Rn, critical points are defined to be the points where the differential of f is a linear map of rank less than min(m, n), i.e. not of maximum rank. This definition immediately extends to maps between smooth manifolds. The image of a critical point under f is 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 differentiable map has measure zero.
