Answer :

The figure is given below:

Let VAB be a given cone of height h, semi-vertical angle α and let x b ethe radius of the base of the cylinder A'B'DC which is inscribed in the cone VAB.
In triangle VO'A',

VO' = x cotα
OO' = VO - VO' = h - x cotα
Let V be the volume of the cylinder. Then,
V = π(O'B')2 (OO')
V = πx2(h - x cotα)
Differentiating with respect to x, we get, 

Now, putting dV/dx = 0, for maxima or minima, we get,

Putting the value of x, we get,

Therefore, there is maxima at x = 2h/3 tanα
Hence, putting the value of x, in formula of volume, we get,

Hence, Proved.

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