(a)[2]
By drawing an appropriate pair of graphs on the same diagram, show that the equation $\ln x = 2e^{-x}$ has exactly one root.
(b)[2]
Verify by calculation that the root is between $1.5$ and $1.6$.
(c)[1]
Show that, if the sequence generated by the iteration formula $x_{n+1} = e^{2e^{-x_n}}$ converges, then its limit is the root of the equation in part (a).
(d)[3]
Use the iterative formula in part (c) to find the root correct to $3$ significant figures. Show the value from each iteration to $5$ significant figures.