Show that the \(x\)-coordinate of \(P\) satisfies the equation
$$4 x ^ { 3 } + 3 x - 3 = 0 .$$
Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.5 and 1 .
Show that the iteration
$$x _ { n + 1 } = \frac { 3 - 4 x _ { n } ^ { 3 } } { 3 }$$
cannot converge to the \(x\)-coordinate of \(P\) whatever starting value is used.
Use the Newton-Raphson method, with initial value 0.5 , to determine the coordinates of \(P\) correct to \(\mathbf { 5 }\) decimal places.