Question

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

Answer 1

There is no maximal surface area.

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

Then, divide the height by 44 and double the radius. The volume does not change, but the area of the circle is 44 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.