 Solution

Use $F$ for the maximized objective function, and adopt optimization constraints $g_i\ge 0$ withThe feasible set is nonempty, since it contains $(0,0)$. It is closed as an intersection of inverse images of closed sets under continuous functions. It lies in the disk of radius $\sqrt{20}$, so is bounded and therefore compact by the Heine-Borel theorem. The objective function is a polynomial, hence continuous. The extreme value theorem now guarantees a global maximizer exists.