4.
$$f ( x ) = x ^ { 2 } + \frac { 5 } { 2 x } - 3 x - 1 , \quad x \neq 0$$
- Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\).
The root \(\alpha\) of the equation \(\mathrm { f } ( x ) = 0\) lies in the interval [0.7, 0.9].
- Taking 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.