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The diagram shows the graph of \(\mathrm { f } ( x ) = \ln ( 3 x + 1 ) - x\), which has a stationary point at \(x = \alpha\). A student wishes to find the non-zero root \(\beta\) of the equation \(\ln ( 3 x + 1 ) - x = 0\) using the Newton-Raphson method.
- (a) Determine the value of \(\alpha\).
(b) Explain why the Newton-Raphson method will fail if \(\alpha\) is used as the initial value. - Show that the Newton-Raphson iterative formula for finding \(\beta\) can be written as
$$x _ { n + 1 } = \frac { 3 x _ { n } - \left( 3 x _ { n } + 1 \right) \ln \left( 3 x _ { n } + 1 \right) } { 2 - 3 x _ { n } } .$$
- Apply the iterative formula in part (ii) with initial value \(x _ { 1 } = 1\) to find the value of \(\beta\) correct to 5 significant figures. You should show the result of each iteration.
- Use a change of sign method to verify that the value of \(\beta\) found in part (iii) is correct to 5 significant figures.