Lagrange Multiplier /w Mixed Inequality/Equality Constraints

[fission14]

Homework Statement

Find the extreme values of the function f(x,y,z) = xy + z^2 in

the set S:= { $$y\geq x, x^2+y^2+z^2=4$$ }

Homework Equations

The Attempt at a Solution

Ok, so This is clearly a lagrange multiplier question. Geometrically, I can see that the region that is the constraint is the surface of the sphere of radius 2, theta in (Pi/4, 5Pi/4) (assuming spherical coords). So it's like half the surface of a sphere.

My problem is that I don't really understand how to set this up. In class we never talked about multiple constraint questions, and it never appeared on our homework. This is my best guess though:

g(x) = x^2+y^2+y^2 -4 , h(x) = x-y <= 0
L(x) = xy+z^2 - \lambda (x^2+y^2+z^2-4) - \mu(x-y)

dL/dx = y-2x\lamdba - \mu =0
dL/dy = x- 2y\lambda + \mu = 0
dL/dz = 2z - 2z\lambda
\lambda(x^2+y^2+z^2-4) =0
\mu(x-y) = 0
x-y <= 0
\mu => 0

This gives me a total of 7 equations to solve. Is this correctly set up? I'm finding it really hard to find information on mixed constraints on google.

 

[fission14]

Ok, i solved those equations as I had them set up, and it looks like I get two extremum,
1 at (0,0,2) (max), and 1 at (sqrt(2), sqrt(2), 0) (min). Just by inspection it looks like this makes sense - and maple also agrees. Success I suppose?