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.
[Math] If the volume of a cylinder is fixed, derive the radius and height that will maximize the surface area
derivativesmaxima-minima
Best Answer
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.