Consider a differentiable function $f$ having domain all positive real numbers, and for which it is known that $f^{\prime}(x) = (4 - x)x^{-3}$ for $x > 0$.
(a) Find the $x$-coordinate of the critical point of $f$. Determine whether the point is a relative maximum, a relative minimum, or neither for the function $f$. Justify your answer.
(b) Find all intervals on which the graph of $f$ is concave down. Justify your answer.
(c) Given that $f(1) = 2$, determine the function $f$.