Problem 12

Extremizing the function

f(x,y,z) = x⁴ + y⁴ + z⁴

on the sphere

g(x,y,z) = x² + y² + z² = 1

leads 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

v-e+f  = 2

which is the famous Euler polyhedra formula.

Questions and comments to knill@math.harvard.edu