 Global maximizer

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.