Maximum and minimum

Maxima and minima are extrema of a real-valued function. A global maximum of $f$ on its domain $D$ is a point $x_{\ast }$ with $f(x)\le f(x_{\ast })$ for every $x\in D$; a global minimum reverses the inequality. A local extremum requires the inequality only in a neighbourhood of the point within $D$. In constrained optimization, $D$ is the feasible set, so an extremum is tested relative to the constraints rather than in every ambient direction. Thus a constrained local maximum or local minimum can occur at a boundary point without being an interior critical point.

A local extremum of a real-valued function is either a local maximum or a local minimum, with comparisons restricted to a neighbourhood in its domain. A differentiable interior local extremum is a stationary point, but the converse can fail, as $f(x)=x^3$ at zero shows.