Answer
Using Theorem 2, we prove that if $f\left( {x,y} \right)$ is differentiable and $\nabla {f_{\left( {x,y} \right)}} = {\bf{0}}$ for all $\left( {x,y} \right)$, then $f$ is constant.
Work Step by Step
If $f\left( {x,y} \right)$ is differentiable and $\nabla {f_{\left( {x,y} \right)}} = {\bf{0}}$ for all $\left( {x,y} \right)$, then by Theorem 2, ${f_x}\left( {x,y} \right)$ and ${f_y}\left( {x,y} \right)$ exist and are continuous. Since $\nabla {f_{\left( {x,y} \right)}} = {\bf{0}}$, so ${f_x} = {f_y} = 0$. Hence, $f$ is constant.
