The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius √3 is:

r² = R² - h²/4
Vc = πr²h = π(R² - h²/4)h = πR²h - πh³/4
dV/dh = πR² - 3πh²/4
(dV/dh)R = √3 = 3π - 3πh²/4
For maximum or minimum volume, dV/dh = 0 ⇒ h = 2
Also, dV/dh changes sign from positive to negative in the neighbourhood of h = 2. Hence, h = 2 is a maximum point.
⇒ r = √2 ⇒ V = 4π cubic units