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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.

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  • $\begingroup$ First, you have to make sure that such a maximum exists. $\endgroup$ – Peter Jul 2 '16 at 10:13
  • $\begingroup$ How to prove so? @Peter $\endgroup$ – ssumukh Jul 2 '16 at 10:14
  • $\begingroup$ 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) $\endgroup$ – Peter Jul 2 '16 at 10:19
  • $\begingroup$ Oh, ok. Thanks :) $\endgroup$ – ssumukh Jul 2 '16 at 10:20
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There is no maximal surface area.

To see that, start with an arbitary cylinder with the given volume.

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.

You can repeat this process as often as you want. It is clear that there is no upper bound for the surface area.

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