extrema

Extrema, in mathematics, are values at which a function attains its largest or smallest values within a given domain. They are classified as maxima or minima, and as global (absolute) or local (relative). A global maximum is a point where f(x) is at least as large as f(x) for every x in the domain; a global minimum is a point where f(x) is at most f(x) for all x. Local extrema satisfy the same inequalities within a neighborhood of the point, and may not be the largest or smallest value over the entire domain.

Critical points are locations where the derivative is zero or undefined (within the domain). Every local extremum must occur at a critical point or at an endpoint of the domain if it is an interval. Conversely, not every critical point is an extremum. In one variable, several tests help identify extrema: the first derivative test compares the sign of f' around a critical point; the second derivative test uses the sign of f'' to conclude a local minimum (f''>0) or a local maximum (f''<0). For a function on a closed interval, endpoints can be global extrema even if they are not critical points.

Existence results include the Extreme Value Theorem: a continuous function on a closed and bounded interval attains both a maximum and a minimum. The concept generalizes to higher dimensions, where extrema occur at points with a vanishing gradient or where the gradient is undefined; second-order tests involve the Hessian matrix.

Extrema have broad applications in optimization problems across mathematics, science, and engineering. Examples: f(x) = (x−2)^2 has a global and local minimum at x = 2; f(x) = −x^2 has a global maximum at x = 0; f(x) = x^3 has no local extrema.