Let $X\in R^3$ and $f(x,y,z) = x^2+3y^2-z^2-xy+2z$, then $df_{(x,y,z)} = (2x-y, 6y-x, 2-2z)$ and the tangent space $TX = \{(a,b,c) \in R^3 | (2x-y)a+(6y-x)b+(2-2z)c=0\}$. Now let $g:R^3\rightarrow R$ s.t. $g((x,y,z))=z$. I am trying to find the critical values of $g$. I know
$dg_{(x,y,z)}:TX_{(x,y,z)}\rightarrow R$
So $dg_{(x,y,z)}$ will be surjective provided that $TX\neq R^2\times \{0\}$, i.e. when $c=0$, because $dg_{(x,y,z)}$ projects the third coordinate. But I am stumped as to how to get a condition on $x,y,z$ such that $dg_{(x,y,z)}$ is not surjective and thus find critical values. A hint is appreciated. Thanks.