There was a previous discussion on this forum about this hypothesis:
If f(x,y,z) is continuous and symmetric, that is, f(x,y,z) = f(y,z,x) = f(z,x,y), then if that function has a global maximum at f(x*,y*,z*), then x*=y*=z*
That discussion didn't include a proof of that, though.
