I agree that the point $(2,2)$, is the only one that meets all criteria. Now we need to compare the value of $f$ at another point on the constraint,...
What I would do is observe that we have cyclical symmetry, that is we may interchange $x_1$ and $x_2$ with no change in either the objective function...
There are online calculators that will spit out the answer to this question, however if you want genuine help, we need to know what you've tried, or...
Head of Mathematics Department, I.E.S. Sta Teresa, Madrid.
Prof. of Mathematical Methods, Univ. Alfonso X El Sabio, Madrid.
(Up until the 2008/2009 academic year. I am now devoted to investigation).