# If the volume of a cylinder is fixed, derive the radius and height that will maximize the surface area

I know how to find the radius and height for minimum surface area. [https://www.physicsforums.com/threads/maximum-surface-area-of-cylinder.332279/ ]. For it to be minimum, h=r/2. It would be great if someone could tell me how to find the values for maximum value.

• First, you have to make sure that such a maximum exists. – Peter Jul 2 '16 at 10:13
• How to prove so? @Peter – ssumukh Jul 2 '16 at 10:14
• Here, we do not have a maximum. In general, it is enough to bound the possible values (as long as you have a continous function, minima and maxima must then exist) – Peter Jul 2 '16 at 10:19
• Oh, ok. Thanks :) – ssumukh Jul 2 '16 at 10:20

Then, divide the height by $4$ and double the radius. The volume does not change, but the area of the circle is $4$ times larger.