Standard optimization problem using Lagrange Multipliers

Hello Everyone!

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

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

Now we can move the constraint and solve this new problem:

$\displaystyle \min \, x+y +\lambda (x^2y^2-4xy)$.

Taking gradients:

$\displaystyle (2\lambda x)y^2+(4\lambda)y+1=0 \, \, \, (1)$

$\displaystyle (2\lambda y)x^2+(4\lambda)x+1=0 \, \, \, (2)$

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

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

$\displaystyle \min_{xy-2 \geq-2} \, x+y$

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

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

Any help is appreciated!