find two positive real numbers x and y such that their product is 800 and x+2y is as small as possible

Answer 1

xy=800, so x == 800/y

x+2y = 800/y+2y
for minimum value, you need

-800/y^2 + 2 = 0
y=20

This is just like the fencing problems where minimum fencing for 3 sides of a rectangle is desired. The amount is evenly divided between the x's and the y's