If the numbers were positive, then AM-GM would have killed it without the need for the second condition

Anyway my solution is tedious, but it works. Let me know if you have a problem somewhere.

We repeatedly use .

let us first put -----------(1)

and then ----------------(2)

Data says

We want to know the maximum value of

But we can use (1) and (2) above to reduce W to

Now set and

Our simplified conditions now read....

This is simple.... :P

The maximum value is 4

(and it occurs for X=Y=2, further you can substitute it above and find the values for which this happens.However the question does not demand the numbers but only the maximum)

I have left a little bit for you, in particular getting . Try it!