Prove that the point is a local maximum if Hessian is null

Question

$f = (x + y + z)^7 - x^6 - y^6 - z^6$. $(0, 0, 0)$ is one of the stationary points and its' Hessian is null. At first I've tried to prove that it's not an extremum, but now I think that it's a local maximum, but I don't know how to prove it formally. Can you give me a hint please?

Answer 1

It is a local maximum. Note that $|(x+y+z)^7| < \max(x^6, y^6, z^6)$ for $(x,y,z)$ sufficiently close to $0$.