Let r be the radius of the base and h the height of a cylinder.

The surface area is given by,

S = 2 π r² + 2 π rh

………(1)

Let V be the volume of the cylinder.

Therefore, V = πr²h

…….Using equation 1

Differentiating both sides w.r.t r, we get,

……….(2)

For maximum or minimum, we have,

⇒

⇒ S = 6πr²

2πr² + 2πrh = 6πr²

h = 2r

Differentiating equation 2, with respect to r to check for maxima and minima, we get,

Hence, V is maximum when h = 2r or h = diameter