 Solution

A $C^1$ embedded submanifold $M\subseteq {\mathbb{R}}^n$ of dimension $n-1$ is locally the graph of a $C^1$ function of $n-1$ coordinates after a change of coordinates. Equivalently, near each point $x_0\in M$ it is a regular level set $\{x:h(x)=0\}$ of a $C^1$ scalar function with $\nabla h(x_0)\ne 0$.

The Lagrange multiplier theorem says that if a $C^1$ function $F$ has a constrained local extremum at $x_0$ on such a level set, thenfor some real $\lambda$. Indeed the derivative of $F$ must vanish on the tangent space $T_{x_0}M=\ker Dh(x_0)$, whose normal space is the line spanned by $\nabla h(x_0)$. The nonzero-gradient condition is essential.