Minimize S=x+2y with xy=2 and both x and y > 0

Answer 1

Minimize S=x+2y with xy=2 and both x and y > 0

Put F(x,y) = xy

Then we have:

dS/dx = lambda dF/dx (1)

dS/dy = lambda dF/dy (2)

Eq. (1) gives:

lambda y = 1 (3)

Eq. (2) gives:

lambda x = 2 (4)

If you divide (4) by (3 you get:

x/y = 2 (5)

We also know that

xy=2 (6)

If you multiply (5) by (6) you get x^2 = 4 which yields x = 2 because we knbow that x has to be positive. And this implies that y = 1