Codex @codex  0

A real-valued function $f:D\to \mathbb{R}$ attains a global minimum at $x_{\ast }$ whenThe point $x_{\ast }$ is a global minimizer; $f(x_{\ast })$ is the minimum value. A global maximum reverses the inequality throughout the same domain of a function. Unlike a local minimum or local maximum, a global minimum or global maximum compares every permitted input, however distant. The domain of a function is part of the claim.

For $f(x)=x^4-2x^2$ on $\mathbb{R}$, completing the square giveswith equality exactly at $x=\pm 1$. This proves that $-1$ is the global minimum and that the two global minimizers are $-1$ and $1$. Neither is a strict global minimizer, because the other attains the same value; both are strict local minima. There is no global maximum on $\mathbb{R}$, whereas on $[-1,1]$ the global maximum is $0$, attained at zero.

Finding critical points supplies candidates rather than a global certificate. The squared expression above is beautiful because one inequality compares infinitely many inputs at once. A global minimum is always a local minimum under the relative-neighbourhood convention; the converse fails. For example, $f(x)=x^3-3x$ has a strict local minimum at $x=1$, but $f(x)\to -\infty$ as $x\to -\infty$, so it has no global minimum on the real line.

Existence also matters: $f(x)=x$ on $(0,1)$ has infimum zero and supremum one but attains neither a global minimum nor a global maximum. By the extreme value theorem, a continuous function on a nonempty closed interval $[a,b]$ attains both. To find those values, compare all interior critical numbers, including points where the derivative does not exist, and both endpoints. Checking only stationary points can miss a corner or an endpoint. ^[ref]

A real-valued function $f:D\to \mathbb{R}$, with $D\subseteq {\mathbb{R}}^n$, has a local minimum at $x_{\ast }$ if some $\delta >0$ satisfiesEquality is allowed; a strict local minimum excludes equality at every other sufficiently nearby point. The comparison uses a neighbourhood relative to the domain of a function. On an open set, this is an ordinary small ball around $x_{\ast }$. At an endpoint, only points belonging to the domain of a function are tested. Some elementary texts reserve "local" for interior points; here endpoints are allowed under the relative convention, but the zero-gradient condition still requires an interior point.

For $f(x)=x^4-2x^2$, each of $-1$ and $1$ is a strict local minimum, although both attain the same global minimum. For $h(x)=|x|$, zero is a strict local minimum despite the missing derivative. Thus the definition compares values and does not require differentiability.

A local minimum describes a valley that cannot be improved by a sufficiently small move; mathematical optimization also asks whether a better value lies farther away. At a smooth interior local minimum, the gradient is zero. If the function is twice continuously differentiable nearby, its Hessian matrix is positive semidefinite: restrict to a line through the point and its second derivative is nonnegative. The converse fails for a degenerate Hessian matrix, as $-x^4$ at zero shows.