We prove the analogous result that was showed for curves about the position of the surface with respect to the tangent plane in terms of its Gauss curvature.
Since the result is local, we suppose that the surface at the point $p$ is tangent to the plane $z=0$ and writes as $z=f(x,y)$ with $p=(q,0)=(0,0,0)$. In such a case, we know that $$K(p)=(f_{xx}f_{yy}-f_{xy}^2)(q).$$
If $K(p)>0$, then the determinant of the Hessian is positive. Since $f_{xx}(q)\not=0$ (on the contrary, $K(p)\leq 0$), then $f_{xx}(q)$ is positive or negative, that is, the Hessian is positive definite or negative definite, respectively. This proves that $q$ is a local minimum or a local maximum, respectively, proving:
Theorem. If $K(p)>0$, then the surface lies in one side of $T_pS$ around $p$.
Corollary. If in any neighbourhood around $p$, $S$ has points in both sides of $T_pS$, then $K(p)\leq 0$.
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