From a solid sphere of mass M and radius R a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is :

When the volume of the cube is maximum, the longest diagonal of cube will be equal to diameter of the sphere.

Let L be the side length of the cube.

FG = GC = L ⇒ FC = √(FG)² + (GC)² = √L² + L² = √2L
⇒ FD = √(FC)² + (CD)² = √(√2L)² + L² = √3L
⇒ √3L = 2R ⇒ L = 2R/√3

Since mass ∝ volume, we have
MC/MS = VC/VS ⇒ MC = (VC/VS) × MS
⇒ MC = (2R/√3)³/((4/3)πR³) × M
⇒ MC = 2M√3/π

And moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is given by
I = (1/6)ML²
⇒ I = (1/6) × (2M√3/π) × (2R/√3)² = 4MR²/9√3π