Consider all right circular cylinders of a fixed lateral surface area (the lateral surface area of a right circular cylinder is given by the formula where is the radius of the base and is the altitude of the cylinder). Prove that the smallest sphere that can be circumscribed about any of them is the sphere with radius , where is the radius of the cylinder.
Proof. The lateral surface area is a constant, and from the picture we see
So, we have the following,
Calling this function we take the derivative,
Then, setting we have,
we have that has a minimum .
Therefore, the value of this minimal radius of the sphere at is given by