extremum

Extremum is a general term for an extreme value of a function, sequence, or other mathematical object. In the common setting of real-valued functions, an extremum refers to a maximum or a minimum. If the value is the largest (respectively smallest) among all values the function takes on its domain, it is a global (absolute) extremum. If the value is the largest or smallest only in a neighborhood of a point, it is a local (or relative) extremum. Extrema may occur at interior points or at the boundary of the domain, and a strict extremum is one where the inequality is strict for nearby points.

In calculus, local extrema are often located via derivatives. If a differentiable function has a local extremum at an interior point, Fermat’s theorem states that the derivative there must be zero (or the derivative does not exist). The second derivative test provides a criterion for classification when the second derivative exists: a local minimum occurs where f'' > 0, a local maximum where f'' < 0; if f'' = 0 the test is inconclusive. For functions on closed intervals, endpoints can host global extrema even when interior critical points do not.

The extreme value theorem gives a core existence result: a continuous function on a compact domain (closed and bounded) attains both a global maximum and a global minimum. If the domain is not compact or the function is not continuous, extrema may fail to exist, though the supremum and infimum can still be defined. In higher dimensions, extrema involve gradients and Hessians, and constrained problems often use methods such as Lagrange multipliers to locate constrained extrema.