Maximapunkter

Maximapunkter are points in the domain of a real-valued function where the function value is at a maximum relative to nearby points (local maximum) or relative to all points in the domain (global maximum). The term distinguishes local from global maxima and strict from non-strict maxima. Formally, x0 is a local maximizer if there exists a neighborhood around x0 such that f(x0) ≥ f(x) for all x in that neighborhood; it is a global maximizer if f(x) ≤ f(x0) for all x in the domain.

In one dimension, if the function f is differentiable, local maxima occur at critical points where f'(x0) = 0 or where the derivative is undefined. The second derivative test states that if f''(x0) < 0, x0 is a local maximum; if f''(x0) > 0, a local minimum; if f''(x0) = 0, the test is inconclusive and other analysis is required. To determine a global maximum, one must compare the values at candidate points with all points in the domain, including endpoints if the domain is finite.

In multiple dimensions, a point x0 is a local maximum if the gradient ∇f(x0) = 0 and the Hessian matrix is negative definite (for a strict local maximum), or negative semidefinite (for a non-strict local maximum). Endpoints and boundary regions can host global maxima, especially on unbounded or non-compact domains.

Maximapunkter are central in optimization, economics, physics and statistics. In data analysis, peaks in curves or time series are often interpreted as local maxima. Algorithms to find maxima include calculus-based methods (gradient ascent) and numerical optimization techniques, which handle non-differentiable or high-dimensional cases. The existence of a global maximum depends on the domain and growth behavior of the function. The term derives from Latin maxima and the Scandinavian word punkt (point).