(a)[2]
On the diagram, sketch the graph of $y = |5x - 4|$, and show that $3 - e^{-\frac{1}{2}x} = |5x - 4|$ has exactly two real roots.
(b)[2]
Show by calculation that $\alpha$ lies between $0.36$ and $0.37$.
(c)[3]
Use the iterative formula $x_{n+1} = \frac{1}{5}\left(7 - e^{-\frac{1}{2}x_n}\right)$ to determine $\beta$ correct to $4$ significant figures. Give the outcome of each iteration to $6$ significant figures.