2
\includegraphics[max width=\textwidth, alt={}, center]{a3e778cb-9f95-4750-ba49-a57ee22af018-2_449_639_388_753}
The diagram shows the curve \(y = x ^ { 4 } + 2 x - 9\). The curve cuts the positive \(x\)-axis at the point \(P\).
- Verify by calculation that the \(x\)-coordinate of \(P\) lies between 1.5 and 1.6.
- Show that the \(x\)-coordinate of \(P\) satisfies the equation
$$x = \sqrt [ 3 ] { \left( \frac { 9 } { x } - 2 \right) }$$
- Use the iterative formula
$$x _ { n + 1 } = \sqrt [ 3 ] { \left( \frac { 9 } { x _ { n } } - 2 \right) }$$
to determine the \(x\)-coordinate of \(P\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.