The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of x from it, is $$I(x)$$. Which one of the graphs represents the variation of $$I(x)$$ with x correctly?

For a solid sphere of mass $$M$$ and radius $$R$$, the moment of inertia about its diameter (center of mass axis) is:

$$I_{cm} = \frac{2}{5}MR^2$$

Using the parallel axis theorem for an axis at a distance $$x$$ from the diameter:

$$I(x) = I_{cm} + Mx^2$$ $$\implies$$ $$I(x) = \frac{2}{5}MR^2 + Mx^2$$

The equation is of the quadratic form $$y = c + a x^2$$, which describes an upward-opening parabola symmetric about the vertical axis.