Theorems from Global extrema

The multivariable extreme-value theorem

An absolute extremum can occur at an interior critical point or anywhere on the boundary $\partial D$. The theorem guarantees existence, but it does not identify the candidates. The boundary is usually an infinite set, so it must be treated as a lower-dimensional optimisation problem.

If either closedness or boundedness fails, an extremum need not be attained; for example, $f(x,y)=x$ on the open unit disk has supremum $1$ but no maximum.