I am trying to find the set of points for which the maximum value of the curvature

$\displaystyle K=\fract{3} { (x^{-2}+y^{-2}+z^{-2})^{-2} }$

occurs for the function $\displaystyle xyz=1$

I have equated the gradients and subject to the restriction of the function xyz=1 i only get one solution the point (1,1,1), since when i solve the equations i get $\displaystyle x^3=y^3=z^3$

but i am not sure if i have done this correctly.Anyway any help would be appreciated.