## Geometry question

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

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

occurs for the function $xyz=1$

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 $x^3=y^3=z^3$

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