Lagrange Multipliers

A method for maximizing or minimizing a general function $f(x, y, z)$ subject to a constraint (or side condition) of the form $g(x, y, z) = k$.

Theorem

Suppose $f,g \in C^1$ and $\nabla g \neq 0$. Then the extremum(s) of $f(x,y)$ subjected to $g(x,y) = k$ are included in each of:

$$\det \frac{\partial (f,g)}{\partial (x,y)} = 0\;\;\text{and}\;\;g(x,y) = 0$$ $$\nabla f(x,y) = \lambda \nabla g(x,y) \;\;\text{and}\;\; g(x,y) = 0$$

When there are $n$ constraint, the system of equations becomes: $$

$$\text{The system of equations is} \begin{cases} \nabla f = \lambda_1 \nabla g_1 + \lambda_2 \nabla g_2 + \cdots + \lambda_n \nabla g_n \\ g_1(x,y) = k_1 \\ g_2(x,y) = k_2 \\ \vdots \\ g_n(x,y) = k_n \end{cases}$$