3. You are given that \(f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30\), for \(x > 2.5\).
- Show that \(f ( x ) = 0\) has a root \(\alpha\) in the interval [3.5, 4].
A student takes 4 as the first approximation to \(\alpha\).
Given \(f ( 4 ) = 3.099\) and \(f ^ { \prime } ( 4 ) = 16.67\) 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 \(f ( x ) = 0\).
[0pt]
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