First, identify any constraints on the variables. For example, any expression inside a square root has to be greater than or equal to 0.

or

and , or ( or ) and ( or )

Let's examine . We know or . Therefore, is false so the only solutions are or .

Now let's examine or . We know . Therefore, and are false so the only solutions are or .

Now we can replace with or and or with or to produce:

and ( or ), or ( or ) and ( or )

From here, we can replace y with -1 and determine for which value of x does the simplified expression have a maximum. Then, we can repeat the replace y with 1 and repeat the procedure. Finally, we can compare those results to replacing x with -1 and 1 and finding for which value of y does this simplified expression have a maximum.

If you're allowed to use calculus, then I know I much more simple method.