8.
$$f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30 , \quad x > 2.5$$
- Show that \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [3.5,4]
A student takes 4 as the first approximation to \(\alpha\).
Given \(\mathrm { f } ( 4 ) = 3.099\) and \(\mathrm { f } ^ { \prime } ( 4 ) = 16.67\) to 4 significant figures, - apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures.
- Show that \(\alpha\) is the only root of \(\mathrm { f } ( x ) = 0\)