f = surface area of a cylinder of height y and radius x: f = 2x² + 2xy.
g = cylinder volume -V = yx² - V.

We get that the partials of f to be 2(2x + y) and  2x; of g to be 2xy and x².
If we divide out the common factors of 2, and and cross multiply, we get: (2x + y)x² = x (2xy) or x² (2x - y) = 0.

Solutions are : y = 2x, and x = 0.

The condition g = 0 implies that y is , so that y goes to infinity as x ^{- 2} as x goes to zero. Thus f goes to infinity as x ^{- 1} as x goes to 0. This means that the critical point at y = 2x is a minimum point for f.

Substituting y = 2x into g tells us that at the minimum point we have .