Lagrange Multiplier Method

Finding a $\lambda$ defined by $\nabla f=\lambda \nabla g$, where $f$ is a function constrained by path $g$.

Method

• Suppose $f$ and $g$ are $C^1$ Class Functions from $U\subset {\mathbb{R}}^n$ to $\mathbb{R}$.
• Let $x_0\in U$ be a point satisfying $g(x_0)=c$ and $\nabla g(x_0)\ne 0$
• Let $S=L_c=\{x:g(x)=c\}$ be Level Set
• If $f|S$ has a Local Extrema at $x_0$ then, $\nabla f(x_0)=\lambda \nabla g(x_0)$ “The gradients are parallel”