Proofs from Local extrema

Fix a coordinate direction and vary only that coordinate through $\mathbf{x}_0$. Since $\mathbf{x}_0$ is an interior local extremum of $f$, the resulting one-variable function also has a local extremum at the corresponding coordinate value. The one-variable Fermat condition makes the associated partial derivative zero. Repeating this for every coordinate gives

which is exactly $\nabla f(\mathbf{x}_0)=\mathbf{0}$.