## FANDOM

186 Pages

Even if Lagrange multiplier is an important mathematical tool, I haven't been used it up to now. It is simple than its look. When we need to optimize two variables with conditioning on the combination of the two variables.

## An Example of Lagrange Multiplier

I describe the contents of video to help readers' understanding.

We want to minimize $C(x,y) = 6x^2 + 12y^2$ subject to $x+y=90$. By Lagrange multiplier, we can write the Lagrange multiplier equation as

$6x^2 + 12y^2 - \lambda(x+y-90)$.

Use the property of Lagrange multiplier that the partial derivatives of the above Lagrange multiplier equation should be equal to zero

$F_x = 12x - \lambda = 0$
$F_y = 24y - \lambda = 0$
$F_\lambda = x + y - 90 = 0$

From the first and the second equations, we will have $x = \lambda/12$ and $y = \lambda/24$. Putting these results to the third equation, we find

$\lambda/12 + \lambda/24 - 90=0$
$\lambda/24 = 30$
$\lambda = 720$,

which result that $x = 30*24/12 = 60$ and $y = 30*12/12 = 30$. Thus, the minimum of $C(x,y)$ becomes $6(60)^2 + 12(30)^2 = 32400$.