Explanation

(b) When a current flows through cylindrical shell, then according to Ampere’s circuital law, magnetic induction inside it will be equal to zero. Hence energy density at $r < R$ is equal to zero
Therefore, (a), (c) and (d) are wrong
When $r > R,\ B = \frac{\mu_{0}i}{2\pi r}$
Since $U = \frac{B^{2}}{2\mu_{0}},$ therefore, outside the shell,
$U = \frac{\mu_{0}i^{2}}{8\pi^{2}r^{2}}$. It means, just outside the shell,
$U = \frac{\mu_{0}i}{8\pi^{2}R^{2}}$
And when $r \rightarrow \infty,\ U \rightarrow 0$
Hence (b) is correct