The implicit function theorem and lagrange multipliers I am reviewing a proof in my multivariable calculus textbook.
After completing a proof of the "Special" Implicit Function Theorem, the case where $F: \Bbb R^{n+1} \rightarrow \Bbb R$ , I read:
Taking a continuously differentiable function $g: \Bbb R^3 \rightarrow \Bbb R$, Consider a level surface $S$ of $g(x,y,z)$,  where $g(x_0,y_0,z_0) = c_0$ and $\frac {\partial f} {\partial z} \neq 0$ , then every vector tangent to $S$ at $(x_0,y_0,z_0)$ is tangent to a curve in $S$. The author then states by the implicit function theorem we need only show this for a graph of the form $z=k(x,y)$. Why is this true?

That's exactly what the Implicit Function Theorem says. Because $\frac{\partial f}{\partial z}\ne0$ (I assume that at $x_0,y_0,z_0)$), there exists a neighbourhood of $(x_0,y_0,z_0)$ where $z=k(x,y)$, with $k$ continuously differentiable, and $g(x,y,k(x,y))=c_0$. 
