6

6. [Maximum mark: 7]

Consider the function $f(x) = \sqrt{x^2 \ln x + 4 - x^2}$, where $x \in \mathbb{R}$, $x > 0$.

(a) Show that the distance, $l$, between the origin and any point on the graph of $f$ is given by $l = \sqrt{x^2 \ln x + 4}$. [1]

(b) Hence, find the $x$‑coordinate of the point on the graph of $f$ which is closest to the origin. [6]