 Solution

The regular level set theorem says that for a $C^1$ map $g:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^r$, if $Dg$ has rank $r$ on $g^{-1}(c)$, then $g^{-1}(c)$ is an $(n-r)$-dimensional submanifold andHere $r=1$ and the given rank condition makes the zero set a two-dimensional submanifold. It is nonempty because $g(0)=f(0)=0$. Part (c) givesConsequentlyThere is also a direct curve proof: $g(p+t(1,1,1))=g(p)=0$, so the entire line through $p$ in this direction lies in $M$. Its velocity at zero is exactly the required tangent vector.