= Global minimum
{title2=$f(x_*)=\min_{x\in D}f(x)$}
= Global minima
{synonym}
A <real-valued function> $f:D\to\mathbb R$ attains a global minimum at $x_*$ when
$$
f(x_*)\leq f(x)\qquad\text{for every }x\in D.
$$
The point $x_*$ is a <global minimizer>; $f(x_*)$ 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 gives
$$
f(x)=(x^2-1)^2-1\geq-1,
$$
with equality exactly at $x=\pm1$. 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. https://openstax.org/books/calculus-volume-1/pages/4-3-maxima-and-minima[Gilbert Strang and Edwin Herman, OpenStax Calculus Volume 1, §4.3: locating absolute extrema]{ref}
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