Shape of Minimum Moment of Inertia for Fixed Volume: How to Prove?

[Grand]

Homework Statement

Find the shape of the solid of revolution with the minimum moment of inertia about it's axis, given the volume is fixed.

Homework Equations

I=\int\frac{1}{2}r^2dm=\int\frac{1}{2}r^2\rho~dV=\int\frac{\rho\pi}{2}r^4dz
V=\int \pir^2dz

I+\lambda V must be stationary.

The Attempt at a Solution

However, when I apply the E-L equs nothing meaningful comes out.

 

[supratim1]

it must be a sphere.

 

[Grand]

How do you prove this. The answer is cylinder.