# Lagrange multiplier theorem

Let $LaTeX: f$ and $LaTeX: h$ be smooth functions and suppose $LaTeX: \nabla h(x^*)$ has full row rank. Then, $LaTeX: x^* \in \arg\max \{f(x): h(x) = 0\}$ only if there exists $LaTeX: v \in \mathbb{R}^m$ such that:
$LaTeX: \nabla f(x*) - v^T \nabla h(x*) = 0.$
The $LaTeX: i$-th component of $LaTeX: v$, $LaTeX: v_i,$ is called the Lagrange multiplier associated with the $LaTeX: i$-th constraint, $LaTeX: h_i(x)=0$. Extensions to remove the rank condition and/or allow inequality constraints were by Carathéodory, John, Karush, and Kuhn & Tucker. Also see the Lagrange Multiplier Rule.