A local max and a local min of a graph function both give non-negative Gaussian curvature

Let $S=\{(x,y,f(x,y))\in\mathbb{R}^3\mid(x,y)\in U\}$ where U is open in $\mathbb{R}^2$ and $f:U\to\mathbb{R}$ is smooth. If f admits a local maximum at $p$ and a local minimum at $q$, then the Gaussian curvatures $k_1$ and $k_2$ to S at the points $(p,f(p))$ and $(q,f(q))$ respectively satisfy: