Kuhn-Tucker

[mathshelpplease]

I am struggling with this problem:



So far I have found that the Lagrangian is
[math]-(x-2)^2-(y-3)^2-\lambda_x(x-2)-\lambda_y(y-2)[/math]
with then the first order conditions of
[math]x:-2(x-2)-\lambda_x=0[/math][math]\lambda_x=-2(x-2)[/math][math]y: -2(y-3)-\lambda_y=0[/math][math]\lambda_y=-2(y-3)[/math]
but am struggling to determine the correct constraints.

I have so far tried
(1) [math]\lambda_x\geq2, x=0[/math](2) [math]\lambda_x=0, x\le2[/math](3) [math]\lambda_y\geq2, y=0[/math](4) [math]\lambda_y=0, y\le2[/math]
but found two possible variations which worked (2 and 4, and 1 and 4) suggesting these to be the wrong constraints.

I then tried
(1) [math]\lambda_x\geq0, x=2[/math](2) [math]\lambda_x=2, x\le0[/math](3) [math]\lambda_y\geq0, y=2[/math](4) [math]\lambda_y=2, y\le0[/math]
but again found both 1 and 3, and 2 and 3 to work.

I was wondering if anyone can be of help in determining the correct constraints?
Thank you.