Let $\Sigma \subset \mathbb{R}^{3}$ be a smooth closed surface. Define the principal curvatures $\kappa_{\max }$ and $\kappa_{\min }$ at a point $p \in \Sigma$. Prove that the Gauss curvature at $p$ is the product of the two principal curvatures.

A point $p \in \Sigma$ is called a parabolic point if at least one of the two principal curvatures vanishes. Suppose $\Pi \subset \mathbb{R}^{3}$ is a plane and $\Sigma$ is tangent to $\Pi$ along a smooth closed curve $C=\Pi \cap \Sigma \subset \Sigma$. Show that $C$ is composed of parabolic points.

Can both principal curvatures vanish at a point of $C$ ? Briefly justify your answer.