Consider the Lagrange Theorem: Let $U \subset \mathbb{R}^{n+k}$ be an open set, $f:U \rightarrow \mathbb{R}$ of class $C^1$, $c \in \mathbb{R}^k$ be a regular value of $g$ and $M = g^{-1}(c)$. Then $$x \ \text{is a critical point of } \ f|_M \Longleftrightarrow \exists\ \lambda \in \mathbb{R}^k \ \text{with}\ \nabla f(x) = \lambda^T \nabla g'(x_0).$$
First: Does anyone has books on the subject? I search for it on Spivak's book and Rudin, but did't find anything. i would appreciate any help!
Second: Consider the function $$\mathcal{L}(x,\lambda) = f(x) -\lambda^T (g(x)-c)$$ i must show that $(x_0,\lambda_0)$ is a critical point of $\mathcal{L}(x,y)$ if, and only if, $x_0$ is a critical point for $f|_{g(x)=c}$. I'm having some troubles to show this. Any hint on this exercise will be very helpful!