extremal

Extremal is a term used across mathematics and the sciences to denote an object that attains an extreme value—either a maximum or a minimum—of a quantity. As an adjective, it describes quantities, points, or objects tied to an extreme; as a noun, an extremal may refer to a function, path, or surface that yields such an extremum.

In calculus and analysis, an extremum is the maximum or minimum of a function on a given domain. Local extrema may differ from a global extremum. Critical points, where the derivative vanishes or is undefined, are candidates for extrema and are classified as minima, maxima, or saddle points after further testing. In the calculus of variations, an extremal is a function or path that makes a functional stationary, typically satisfying the Euler–Lagrange equations.

In geometry and complex analysis, extremal concepts include extremal length, a conformal invariant associated with a family of curves, and extremal functions that optimize certain bounds under constraints. These ideas often arise in optimization problems constrained by geometry or other global conditions.

In physics and related disciplines, extremal configurations frequently correspond to states that minimize energy or maximize other quantities under given constraints. Examples include extremal black holes in general relativity, which saturate certain charge-to-mass bounds and exhibit distinctive thermodynamic properties.

Overall, the notion of extremal centers on achieving the boundary value of interest within a defined setting, whether for a simple function or a complex functional under multiple constraints.