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thumb|300px|right|LaGrange Multipliers

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 $.