Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to 2/3 of the diameter of the sphere.

Let the radius and height of cone be r and h respectively

Radius of sphere = R

R² = r² + (h - R)²

R² = r² + h² + R² - 2hR

r² = 2hR - h² …1

Assuming volume of cone be V

Volume of cone, (from equation 1)

Condition for maxima and minima is

⇒ 4hR - 3 h²= 0

For , < 0

V will be maximum for