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 $1.\nabla g=\vec{0}\text{and}g(x,y)=K$ $2.\nabla f=\lambda \nabla g\text{and}g(x,y)=K\text{for some constant }\lambda$ $3.\text{Plug all points from (1) and (2) into }f(x,y)$

$\lambda$ 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.