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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 is 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.

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. Its 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}.

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.

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.

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.

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.

• 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
• Critical point of complex quadratic polynomial