Hello Everyone!

I'm a bit stuck on this one, and was wondering whether you could give me a hint to go about it:

\min_{(xy-2)^2 \leq4} \, x+y.

Now we can move the constraint and solve this new problem:
\min \, x+y +\lambda (x^2y^2-4xy).

Taking gradients:
(2\lambda x)y^2+(4\lambda)y+1=0 \, \, \, (1)
(2\lambda y)x^2+(4\lambda)x+1=0 \, \, \, (2)

Now, I know I need to find x and y in terms of \lambda, something that would yield one variable in the terms of another and \lambda which made me throw that whole approach and think about these two problem:

\min_{xy-2 \leq2} \, x+y, and
\min_{xy-2 \geq-2} \, x+y

Both problems would yield that (x,y)=(\pm 2, \pm 2).

I would find the minimum in each case, then rule out the greater of these two minimums. Would this work?

Any help is appreciated!