Let $U \subset \mathbb{R}^{2}$ be an open set. Let $\Sigma \subset \mathbb{R}^{3}$ be a surface locally given as the graph of an infinitely-differentiable function $f: U \rightarrow \mathbb{R}$. Compute the Gaussian curvature of $\Sigma$ in terms of $f$.

Deduce that if $\widehat{\Sigma} \subset \mathbb{R}^{3}$ is a compact surface without boundary, its Gaussian curvature is not everywhere negative.

Give, with brief justification, a compact surface $\widehat{\Sigma} \subset \mathbb{R}^{3}$ without boundary whose Gaussian curvature must change sign.