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 the derivative is equal to zero or does not exist. It is also called a stationary point.
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[edit] Introduction
For a 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) and a minimum occurs when all eigenvalues are positive (index zero). Morse theory studies both finite and infinite dimensional manifolds using these ideas.
[edit] Used to find maxima and minima
In one or several variables, the maxima and minima of a function (if they exist) can occur either at its critical points or at points on its boundary, or points where the function is not differentiable.
A critical point is sometimes not a local maximum or minimum. In that case it is called a saddle point.
[edit] Alternative definition
Critical points are also sometimes defined to be points where the derivative of a function is not of maximum rank, i.e. where it fails to be a submersion. The value of a function at a critical point, if defined, is called a critical value. Sard's theorem states that the set of critical values, in this sense of critical point, of a differentiable function has measure zero.
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.
