Find and solve the FOC associated to the problem: Min xy subject to x^2+ 4y^2= 1.

spaul

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1. Find and solve the first order conditions associated to the problem: Minimize xy subject to x2 + 4y2 = 1.

Need to solve this using the Lagrangian

I have done the following

U(x,y) - λ(x^2+4y^2-1)

(1) dL/dx = y-λ2x=0
(2) dL/dy = x- λ8y=0
(3) dL/d
λ = -(x^2 + 4y^2-1) = 0

(dL/dx)/(dL/dy)= (y-
λ2x)/(x- λ8y)

x/y=8y/2x
(x/y)2x=8y
2x^2/y=8y
2x^2/8y=y


I just wanted to make sure these steps were right before i substituted it in to the original formula.

Thanks
;)





 

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They are correct but awkward. Why in the world would you keep the "8" and "2" throughout? With \(\displaystyle \frac{8x}{y}= \frac{2y}{x}\) the first thing I would do is divide both sides by 2: \(\displaystyle \frac{4x}{y}= \frac{y}{x}\). And then get rid of the fractions by multiplying both side by xy: \(\displaystyle 4x^2= y^2\) so that y can be 2x or -2x.
 
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