= Solution
Use $F$ for the maximized <objective function>, and adopt <optimization constraints> $g_i\geq0$ with
$$
g_1=2-x_1,\qquad g_2=20-x_1^2-x_2^2,\qquad g_3=x_2-x_1^2.
$$
The <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 \b[a global maximizer exists].
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