Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

Let 'H' be the height of the cone, 'R' be the radius of the cone, 'h' be the height of the cylinder, 'r' be the radius of the cylinder and 'S' be the lateral surface area of the cylinder. To prove - r = R/2. Radius of cylinder, r = R(1 - h/H) Now since S = 2πrh, put r = R(1 - h/H) in S = 2πrh as shown below: S = 2π(R(1 - h/H))h Differentiating with respect to h we get, dS/dh = 2π(RH - Rh)/H Again differentiating with respect to h that is d²S/dh² = -2π(R/H) Now put dS/dh = 0 to find the stationary points: RH - 2Rh = 0 Implies h = H/2 Now consider, r = R(1 - h/H) Put h = H/2 in r = R(1 - h/H), we get r = R/2.