Extrémum

Extrémum (also extremum) is a mathematical term for a maximum or minimum value attained by a function on its domain. The input at which this value occurs is typically called a point of extremum, while the value itself is an extremum value. Extrema can be local (relative) or global (absolute). A local maximum is a point where f(x0) is at least as large as nearby values; a local minimum is at most as large as nearby values. A global maximum (absolute maximum) is the largest value of f on the entire domain, and a global minimum (absolute minimum) is the smallest.

To find local extrema of a differentiable function f, one looks at critical points where the derivative vanishes or is undefined. In one variable, the first derivative test classifies the type; the second derivative test can confirm a local minimum when f''(x0) > 0 and a local maximum when f''(x0) < 0. In several variables, one uses the gradient: critical points satisfy ∇f = 0; the Hessian matrix's definiteness determines the nature of the extrema.

Extrema on constrained domains require different methods, such as Lagrange multipliers. If the domain is closed and bounded and f is continuous, the extreme value theorem guarantees at least one global maximum and one global minimum on the domain.

Non-smooth cases: extrema can occur at points where derivatives do not exist, for example at corners or cusps (e.g., |x| has a global minimum at x = 0).

Applications span optimization, economics, physics, and engineering, where extrema represent optimal performance, cost, or energy.