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 $.