# Method of Lagrange

We use the Method of Lagrange to find extremums of multivariate functions with respect to a constraint. To do so, we first find its critical point (with respect to a constraint), and afterwards, evaluating those points using $f(x,y)$, we conclude

- Biggest values = global max
- smallest values = global min

Abstract

To find the critical points of $f(x,y)$ subject to a constraint $g(x,y)=K$ where ($K$ is a constant), find the values of $x$ and $y$ for which $1.āg=0andg(x,y)=K$ $2.āf=Ī»āgandg(x,y)=KforĀ someĀ constantĀĪ»$ $3.PlugĀ allĀ pointsĀ fromĀ (1)Ā andĀ (2)Ā intoĀf(x,y)$

$Ī»$ is known as the *Lagrange Multiplier*.

Exceptions

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### Example

### Inequality

For inequalities, find critical points of $f$, then only consider the points inside the boundary condition.

Then, use lagrange to check at the edge of the boundary.