Proofs from Constrained extrema

At a regular point of $C$, at least one of $g_x,g_y$ is nonzero, so the constraint is locally a smooth curve. A tangent vector $\mathbf{v}$ to that curve satisfies

because $g$ is constant along the constraint. At a constrained extremum, the first-order change of $f$ must also vanish for every feasible tangent vector:

In the plane, both gradients are therefore normal to the same tangent line. Since $\nabla g\ne\mathbf{0}$, the two normals are parallel, so $\nabla f=\lambda\nabla g$ for some scalar $\lambda$. The existence of maximum and minimum values follows by applying the extreme-value theorem to the closed and bounded constraint curve.