Global maximizer ($x_{\ast }$)

A global maximizer of $f:X\to \mathbb{R}$ is a point $x_{\ast }\in X$ satisfying $f(x_{\ast })\ge f(x)$ for every $x\in X$. The global maximum is the attained value $f(x_{\ast })$; argmax collects all points attaining that value:The set can be empty or contain many points. For $f(x)=\cos x$ on the real line, the global maximum is one and the global maximizers are $2\pi k$, $k\in \mathbb{Z}$. No single representative is specified by the optimization problem.

A stationary point is a different kind of condition. For a differentiable function on an open real interval, an interior global maximizer has zero derivative, but the converse fails: $f(x)=x^3$ has zero derivative at zero without a local maximum. A boundary global maximizer need not have zero derivative either; $f(x)=x$ on $[0,1]$ attains its largest value at the right endpoint.

The extreme value theorem supplies a useful sufficient existence condition: a continuous function on a nonempty compact set attains its maximum. Compactness is sufficient, rather than necessary; the cosine example attains a global maximum on the noncompact real line.

The argmax of a function is the set of inputs where its largest value is attained:It records global maximizers, whereas ${\max }_{x\in X}f(x)$ records the value. For $f(x)=-(x-2{)}^2$, the argmax on $\mathbb{R}$ is $\{2\}$ and the maximum value is zero. If $f$ is constant, every input belongs to the argmax.

The argmax can be empty: $f(x)=x$ on $(0,1)$ has supremum one but never attains it. A continuous function on a nonempty compact set does attain its maximum by the extreme value theorem. Writing a single input as an argmax therefore implicitly assumes uniqueness or specifies a rule for choosing between ties. This distinction matters when an optimization problem is used to define an estimator.