Critical point

A function has a local maximum at $x_0$ if $f(x)\le f(x_0)$ throughout some neighbourhood of $x_0$. The inequality need not be strict. Negating the function exchanges local maxima and local minima.

A function has a strict local maximum at $t$ if some neighbourhood contains no other point $s$ with $f(s)\ge f(t)$. Unlike a non-strict local maximum, equality at another sufficiently nearby point is excluded. Use neighbourhoods relative to the domain, including when $t$ is an endpoint.

A stationary point is nonisolated if every neighbourhood contains other stationary points. For a twice differentiable function, a regular smooth curve of stationary points through a point forces its Hessian matrix to annihilate the curve's nonzero tangent: differentiate the vanishing gradient along the curve. Thus the Hessian matrix has a zero eigenvalue there, but that alone need not determine whether the point is a local minimum or local maximum.

The critical set of a differentiable map consists of the points where its derivative fails to have full rank.

A function has a local minimum at $x_0$ when $f(x)\ge f(x_0)$ throughout some neighbourhood of $x_0$. For a twice differentiable function, its Hessian matrix at an interior local minimum is positive semidefinite.

A global minimum of a function on a domain is a point whose function value is no greater than the value at any other point of that domain.

A global minimizer is a point attaining the infimum of an objective. For an extended-real functional, it is usual to require a finite objective value; this excludes points outside its effective domain. A convention allowing $+\infty$ as the attained value gives every point as a formal minimizer of the identically infinite function.