5 The equation
$$x ^ { 3 } - 5 x + 3 = 0$$
may be solved by the Newton-Raphson method. Successive approximations to a root are denoted by \(x _ { 1 } , x _ { 2 } , \ldots , x _ { n } , \ldots\).
- Show that the Newton-Raphson formula can be written in the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), where
$$\mathrm { F } ( x ) = \frac { 2 x ^ { 3 } - 3 } { 3 x ^ { 2 } - 5 }$$
- Find \(\mathrm { F } ^ { \prime } ( x )\) and hence verify that \(\mathrm { F } ^ { \prime } ( \alpha ) = 0\), where \(\alpha\) is any one of the roots of equation (A).
- Use the Newton-Raphson method to find the root of equation (A) which is close to 2 . Write down sufficient approximations to find the root correct to 4 decimal places.