The Lagrangian is
where we assume , with critical points wherever the partial derivatives are identically zero:
(note that the latter case requires , and we'll see the same requirement in the next two equations)
If either , , or , then we can throw out the latter cases for , , and , since we don't want . If, for instance, we pick , then we're left with , for which . This means we have 6 critical points where two of the coordinates are 0, and the remaining one is either 1 or -1.
In the latter case,
If , then
A similar thing happens in all the other cases, which leads us to getting 8 possible critical points, ; at each of these points, we have .
To summarize: has a maximum value of 1/27 and a minimum value of 0.