A cylinder of radius x is inscribed in a fixed sphere of internal radius R, the circumferences of

the circular ends of the cylinder being in contact with the inner surface of the sphere. Show

that A²=16\pi ²x²(R²-x²), where A is the curved surface area of the cylinder.

Show that, if x is made to vary, the maximum value of A is obtained when the height of

the cylinder is equal to its diameter.