(a)[3]
Find an expression for $\frac{dy}{dx}$ and show that the $x$-coordinate of $M$ satisfies the equation $x = \frac{3x + 2}{3 \ln x}$.
(b)[2]
Use the equation from part (a) to demonstrate by calculation that the $x$-coordinate of $M$ lies between $3$ and $4$.
(c)[3]
Use an iterative formula, based on the equation in part (a), to determine the $x$-coordinate of $M$ correct to $5$ significant figures. Give each iteration result to $7$ significant figures.