Critical point (mathematics)
[2] Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not holomorphic).
[3][4] Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined).
In other words, the critical points are those where the implicit function theorem does not apply.
A critical point of a function of a single real variable, f (x), is a value x0 in the domain of f where f is not differentiable or its derivative is 0 (i.e.
Although it is easily visualized on the graph (which is a curve), the notion of critical point of a function must not be confused with the notion of critical point, in some direction, of a curve (see below for a detailed definition).
If g(x, y) is a differentiable function of two variables, then g(x,y) = 0 is the implicit equation of a curve.
This means that the tangent of the curve is parallel to the y-axis, and that, at this point, g does not define an implicit function from x to y (see implicit function theorem).
Such a critical point is also called a bifurcation point, as, generally, when x varies, there are two branches of the curve on a side of x0 and zero on the other side.
If f is not differentiable at x0 due to the tangent becoming parallel to the y-axis, then x0 is again a critical point of f, but now (x0, y0) is a critical point of its graph for the projection parallel to the y-axis.
If one considers the upper half circle as the graph of the function
Sendov's conjecture asserts that, if all of a function's roots lie in the unit disk in the complex plane, then there is at least one critical point within unit distance of any given root.
Critical points play an important role in the study of plane curves defined by implicit equations, in particular for sketching them and determining their topology.
In fact it is the specialization to a simple case of the general notion of critical point given below.
, where f is a differentiable function of two variables, commonly a bivariate polynomial.
When the curve C is algebraic, that is when it is defined by a bivariate polynomial f, then the discriminant is a useful tool to compute the critical points.
If the function is at least twice continuously differentiable the different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.
A critical point at which the Hessian matrix is nonsingular is said to be nondegenerate, and the signs of the eigenvalues of the Hessian determine the local behavior of the function.
In the case of a function of a single variable, the Hessian is simply the second derivative, viewed as a 1×1-matrix, which is nonsingular if and only if it is not zero.
For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point.
A non-degenerate critical point is a local maximum if and only if the index is n, or, equivalently, if the Hessian matrix is negative definite; it is a local minimum if the index is zero, or, equivalently, if the Hessian matrix is positive definite.
By Fermat's theorem, all local maxima and minima of a continuous function occur at critical points.
Therefore, to find the local maxima and minima of a differentiable function, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.
This requires the solution of a system of equations, which can be a difficult task.
When the function to minimize is a multivariate polynomial, the critical points and the critical values are solutions of a system of polynomial equations, and modern algorithms for solving such systems provide competitive certified methods for finding the global minimum.
Sard's theorem states that the set of critical values of a smooth map has measure zero.
be a differential map between two manifolds V and W of respective dimensions m and n. In the neighborhood of a point p of V and of f (p), charts are diffeomorphisms
This definition does not depend on the choice of the charts because the transitions maps being diffeomorphisms, their Jacobian matrices are invertible and multiplying by them does not modify the rank of the Jacobian matrix of
If M is a Hilbert manifold (not necessarily finite dimensional) and f is a real-valued function then we say that p is a critical point of f if f is not a submersion at p.[8] Critical points are fundamental for studying the topology of manifolds and real algebraic varieties.
The link between critical points and topology already appears at a lower level of abstraction.
In the case of real algebraic varieties, this observation associated with Bézout's theorem allows us to bound the number of connected components by a function of the degrees of the polynomials that define the variety.
