Farmer knew Lagrangian optimization!

Problem:



Solution:

Minimize x + 2y subject to xy = 405000

The Lagrangian is

$ L(x, y, \lambda) = x + 2y + \lambda(xy - 405000) $

The partial derivatives must be 0, so we get

$ 1 + \lambda y = 0 $
$ 2 + \lambda x = 0 $

Multiply the first equation by $ x $ and multiply the second equation by $ y $ gives $ x - 2y = 0 $.

In other words, $ x = 2y $, so $ xy = 2y^2 = 405000 $, $ y = 450, x = 900 $.