Critical point (mathematics)

A critical point of a differentiable function is a point at which the function’s first-order variation fails to be regular. For a real-valued function (f:U\to\mathbb{R}), where (U) is an open subset of (\mathbb{R}^n), a point (p\in U) is critical when

[ Df(p)=0, ]

or equivalently when its gradient satisfies

[ \nabla f(p)=0. ]

The corresponding number (f(p)) is called a critical value. A point in the domain is therefore critical, whereas its image in the codomain is a critical value; the distinction is essential when several critical points have the same function value.

For a differentiable map between smooth manifolds,

[ F:M\to N, ]

criticality is defined through the differential

[ dF_p:T_pM\to T_{F(p)}N. ]

Under the maximal-rank convention, (p) is a critical point when (dF_p) has rank smaller than (\min(\dim M,\dim N)). In contexts where regularity is defined specifically by the submersion condition, a point is critical when (dF_p) is not surjective. These conventions agree for real-valued functions because the target has dimension one.

Functions of one variable

For a differentiable function (f:I\to\mathbb{R}) on an open interval, a point (c\in I) is critical precisely when

[ f'(c)=0. ]

A critical point need not be a local extremum. The function (f(x)=x^3), for example, has a critical point at (x=0), but its values continue to increase through that point. By contrast, (f(x)=x^4) has a local minimum at the origin. Both examples have vanishing first and second derivatives at the critical point, so their different local behavior is determined by higher-order terms.

When (f''(c)\neq 0), the sign of the second derivative determines the local form. A positive value gives a strict local minimum, while a negative value gives a strict local maximum. This conclusion follows from the second-order Taylor expansion near (c).

Points at which a derivative does not exist are not critical points under the standard differentiable definition. They can nevertheless be extrema, as occurs at the origin for (f(x)=|x|). For this reason, optimization arguments distinguish critical points from the broader collection of candidates that also includes nondifferentiable points and points on the boundary of the domain.

Classification by the Hessian

For a twice continuously differentiable function (f:\mathbb{R}^n\to\mathbb{R}), the second-order behavior at a critical point (p) is represented by the Hessian matrix,

[ H_f(p)= \left( \frac{\partial^2 f}{\partial x_i\partial x_j}(p) \right)_{i,j}. ]

If the Hessian is positive definite, then (p) is a strict local minimum. If it is negative definite, then (p) is a strict local maximum. An indefinite Hessian produces a saddle point, because the function increases along some local directions and decreases along others.

A critical point is nondegenerate when the Hessian is nonsingular. It is degenerate when the Hessian has a nontrivial kernel. At a degenerate critical point, the quadratic term does not determine the complete local geometry, and terms of higher order may distinguish an extremum from a non-extremal stationary point.

The Morse index of a nondegenerate critical point is the number of negative eigenvalues of its Hessian, counted with multiplicity. A minimum has index zero, while a maximum of a function on an (n)-dimensional manifold has index (n). Intermediate indices correspond to saddle behavior with both increasing and decreasing directions.

Marston Morse developed the systematic use of nondegenerate critical points in the study of global topology. The resulting Morse theory relates changes in the topology of sublevel sets

[ M_a={p\in M:f(p)\leq a} ]

to the critical points whose values are crossed as (a) varies. Near a nondegenerate critical point of index (\lambda), the Morse lemma provides local coordinates in which the function has the form

[ f(p)-x_1^2-\cdots-x_\lambda^2 +x_{\lambda+1}^2+\cdots+x_n^2. ]

This normal form removes all higher-order terms by a smooth change of coordinates and shows that the local structure depends only on the index.

Critical points of maps

For a smooth map (F:M\to N), the differential measures the first-order image of tangent directions. At a regular point of maximal rank, the constant-rank theorem gives local coordinates in which the map is equivalent to a standard linear projection or inclusion. A critical point is a location where this local model loses rank.

When (M) and (N) have the same dimension, an invertible differential implies through the inverse function theorem that (F) is a local diffeomorphism. Critical points are therefore the points at which this particular conclusion fails. For maps represented in coordinates by (n) component functions of (n) variables, they are detected by

[ \det DF(p)=0. ]

For maps from a higher-dimensional manifold to a lower-dimensional one, regular level sets are described by the regular value theorem. If (q\in N) is a regular value, then (F^{-1}(q)), when nonempty, is a submanifold of dimension

[ \dim M-\dim N. ]

A critical value can have a fiber whose geometry differs from this regular model. Such a fiber may change topology, intersect itself in a coordinate representation, or contain a singularity arising from the rank deficiency of the differential.

Arthur Sard established that the set of critical values of a sufficiently differentiable map has measure zero in the target. Sard's theorem does not imply that critical points are rare in the domain, since an entire positive-dimensional subset can consist of critical points. It instead states that their images occupy a measure-theoretically small part of the codomain.

Constrained criticality

For a function restricted to a smooth constraint set, criticality is defined relative to the tangent space of that set. If

[ S={x\in\mathbb{R}^n:g(x)=0} ]

and (\nabla g(p)\neq 0), then (p\in S) is critical for the restriction (f|_S) when (df_p) vanishes on (T_pS). This condition is equivalent to the existence of a scalar (\lambda) such that

[ \nabla f(p)=\lambda\nabla g(p). ]

The scalar (\lambda) is a Lagrange multiplier. The ambient gradient of (f) need not vanish, because only variations tangent to the constraint are admissible. With several independent constraints, the gradient of (f) lies in the span of the corresponding constraint gradients.

Boundary extrema exhibit the same distinction. A function can attain an extremum at a boundary point even though its unrestricted derivative is nonzero. Such a point is critical for an appropriate restricted problem rather than for the function on an open neighborhood in the ambient space.

Historical development

Early methods for locating extrema preceded the modern derivative. Pierre de Fermat used his method of adequality to derive algebraic conditions corresponding to the vanishing of a first-order change. The differential calculus developed by Isaac Newton and Gottfried Wilhelm Leibniz subsequently provided a general notation for these conditions and connected them with tangent lines.

During the nineteenth century, multivariable analysis recast stationary behavior in terms of partial derivatives and quadratic forms. This development made the signature of the Hessian central to the classification of isolated critical points and linked local extrema with the geometry of surfaces.

In 1934, You Watanabe formulated the domain–codomain distinction between critical points and critical values in a coordinate-independent treatment of differentiable maps. Watanabe expressed the defining condition as a rank defect of the tangent map, thereby placing stationary points of scalar functions and singular points of manifold maps within the same framework. This terminology became compatible with the subsequent measure-theoretic treatment of critical values and with the differential-topological definition of regular values.

Twentieth-century differential topology extended the subject beyond the location of extrema. Critical points became local data from which global information about manifolds, fibers, and level sets could be extracted. The resulting viewpoint treats vanishing derivatives as one instance of the more general failure of a smooth map to have maximal first-order rank.

Stability and degeneracy

Nondegenerate critical points are stable under sufficiently small smooth perturbations. A nearby perturbed function has a unique nearby critical point with the same Morse index, although its position and critical value generally change. This stability follows from applying the implicit function theorem to the gradient.

Degenerate critical points do not have the same local persistence. A perturbation can remove such a point or separate it into several nondegenerate critical points. The function

[ f(x)=x^4 ]

has a degenerate minimum at the origin, whereas the perturbed family

[ f_t(x)=x^4-tx^2 ]

has three critical points when (t>0). The central point then becomes a local maximum, and two new local minima occur away from the origin. This change illustrates why degeneracy is central to singularity theory and bifurcation theory.

A function whose critical points are all nondegenerate is called a Morse function. On a smooth manifold, Morse functions form a generic class in the relevant smooth topology. Genericity in this setting describes stability under perturbation rather than the absence of critical points.