The circular cylinder is inscribed in a sphere.

$\"\"$

Observe the figure.

$\"\"$ be the radius of the sphere.

$\"\"$ be the radius of the cylinder.

From pythagorean theorem,

$\"\"$

The volume of the circular cylinder is $\"\"$ .

$\"\"$

Apply derivative on each side withj respect to $\"\"$ .

$\"\"$

The volume of the cylinder is maximum when its first derivative becomes zero.

Equate $\"\"$ to zero.

$\"\"$

Therefore the height of the cylinder is $\"\"$ .

Substitute $\"\"$ in $\"\"$ .

$\"\"$

Therefore the largest volume of the cylinder is $\"\"$ .

The largest volume of the cylinder is $\"\"$ .