Let r and h be the radius and height of the cone respectively inscribed in a sphere of radius R.

Let V be the volume of cone.

Then, V=

And height of cone h = R +

Now,

Now, if,

After solving this we get,

So, when, , then < 0

Then, by second derivative test, the volume of the cone is the maximum when

So, when, , h = R +

Therefore, V =

Therefore, the volume of the largest cone that can be inscribed in the sphere is the volume of the sphere.