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

Let VAB be the cone of base of radius , r = OA and height , h = VO.

Let a cylinder of base of radius OC = x be inscribed in the cone.

From the figure , it is clear that ΔVOB∼ΔB'DB

Let S be the curved surface area of cylinder.

Then, S= 2(OC) (B'D)

For maximum  or minimum values of S, we must have

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