The closure is closed and bounded in $\mathbb R^2$, hence <compact> by the <Heine-Borel theorem>. It is nonempty. The <extreme value theorem> therefore gives a point of the closure at which the continuous function attains its maximum. The closure, rather than the open domain, is essential.

Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 1 / 2I / i / Solution

 Solution (source code)