6 It is given that the equation \(3 x ^ { 3 } + 5 x ^ { 2 } - x - 1 = 0\) has three roots, one of which is positive.
- Show that the Newton-Raphson iterative formula for finding this root can be written
$$x _ { n + 1 } = \frac { 6 x _ { n } ^ { 3 } + 5 x _ { n } ^ { 2 } + 1 } { 9 x _ { n } ^ { 2 } + 10 x _ { n } - 1 } .$$
- A sequence of iterates \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) which will find the positive root is such that the magnitude of the error in \(x _ { 2 }\) is greater than the magnitude of the error in \(x _ { 1 }\). On the graph given in the Printed Answer Book, mark a possible position for \(x _ { 1 }\).
- Apply the iterative formula in part (i) when the initial value is \(x _ { 1 } = - 1\). Describe the behaviour of the iterative sequence, illustrating your answer on the graph given in the Printed Answer Book.
- A sequence of approximations to the positive root is given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\). Successive differences \(x _ { r } - x _ { r - 1 } = d _ { r }\), where \(r \geqslant 2\), are such that \(d _ { r } \approx k \left( d _ { r - 1 } \right) ^ { 2 }\) where \(k\) is a constant. Show that \(d _ { 4 } \approx \frac { d _ { 3 } ^ { 3 } } { d _ { 2 } ^ { 2 } }\) and demonstrate this numerically when \(x _ { 1 } = 1\).
- Find the value of the positive root correct to 5 decimal places.