Theorem

Assume $f:\mathbb{R}^{k}\to \mathbb{R}$ and $g:\mathbb{R}^{k}\to \mathbb{R}$ are continuously differentiable functions. We want to solve the following constrained minimization problem (CMP): $$\begin{align*} &\text{minimize }f(\mathbf{x})\\ &\text{subject to }g(\mathbf{x})=0 \end{align*}$$Let $\nabla u=\left( \frac{ \partial }{ \partial x_{1} }{u}\dots,\frac{ \partial }{ \partial x_{k} } u \right)$ denote the gradient of a $u:\mathbb{R}^{k}\to \mathbb{R}$. We say now if $\mathbf{x}^{*}$ is a solution of the CMP and $\nabla g(\mathbf{x}^{*})\not=0$, then $\exists\lambda\in\mathbb{R}^{*}$ called the Lagrange multiplier such that $$\nabla f(\mathbf{x}^{*})+\lambda \nabla g(\mathbf{x}^{*})=0$$ ## Remark The theorem yields the following equations: $$\begin{align*} \left.\frac{ \partial }{ \partial x_{j} } (f(\mathbf{x})+\lambda g(\mathbf{x}))\right|_{\mathbf{x}=\mathbf{x}^{*}}&=0\\ g(\mathbf{x}^{*})&=0 \end{align*}$$this gives us $k+1$ equations for $k+1$ unknowns.