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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 is equal to zero or does not exist: it is points that are either stationary points or non-differentiable points.

★ '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'.

Contents

Versus stationary point

Optimization

Several variables

Gradient vector field

Alternative definition (not full rank)

See also

Versus stationary point

The term "critical point" is often confused with "stationary point". Critical point is more general: a critical point is either a stationary point or the derivative is not defined there.
For a smooth function, these are interchangeable, hence the confusion; when a function is understood to be smooth, one can refer to stationary points as critical points, but when it may be non-differentiable, one should distinguish these notions.
In one dimension, critical point generally means "possibly non-differentiable", as in calculus.
In higher dimensions, critical point generally means "derivative is zero" (and the function is understood to be smooth), as in Morse theory.

Optimization

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

★ Singular point of an algebraic variety

★ Singular point of a curve

★ Van Hove singularity

★ Fermat's theorem

★ Inflection point

★ Saddle point

★ Ridge detection

★ Sard's lemma