
Extremizing the function
f(x,y,z) = x^{4} + y^{4} + z^{4}on the sphere g(x,y,z) = x^{2} + y^{2} + z^{2} = 1leads to many critical points: there are 26. For every vertex of the cube (v=8), for every face (f=6) and for every edge (e=12), there is a critical point. Now 8+6+12=26. Note that like for any polyhedron, one could have counted the edges with the formula ve+f = 2which is the famous Euler polyhedra formula. 