Minimumpunkt

Minimizedpunkt is the point at which a real-valued function attains its minimum value on a given domain. It can be local or global; a local minimumpunkt is where the function value is not exceeded in a small neighborhood, while a global minimumpunkt is the smallest value achieved over the entire domain.

In one variable: if a function f is differentiable, a point x is a local minimumpunkt if f'(x) = 0 and f''(x) > 0, or if the derivative changes sign from negative to positive. Endpoints of the domain must be considered, since a minimum can occur there even if f'(x) ≠ 0.

In several variables: a point x is a local minimumpunkt if the gradient ∇f(x) = 0 and the Hessian matrix at x is positive definite. If the Hessian is indefinite, the point is typically a saddle point. A positive semidefinite Hessian may be inconclusive, and additional analysis may be required.

Global minima exist under certain conditions: if the domain is compact and the function is continuous, a global minimum is guaranteed to exist. If the function is convex on a convex domain, any local minimum is also a global minimum, which simplifies analysis and computation in optimization problems.

Applications and methods: minimization problems arise in mathematics, physics, economics, and computer science. They are tackled with calculus-based techniques or numerical algorithms such as gradient descent, Newton’s method, and various forms of linear or nonlinear programming, including constrained optimization where the feasible set imposes additional requirements.

Example: the function f(x) = (x − 2)^2 + 3 has a minimumpunkt at x = 2 with minimum value 3.