Multiple roots and starting value selection

5
\includegraphics[max width=\textwidth, alt={}, center]{b9f29713-bc86-4869-9e54-195208e5e81d-3_661_734_267_703} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 4.36$$ The curve has turning points at \(x = 1\) and \(x = 2\) and crosses the \(x\)-axis at \(x = \alpha , x = \beta\) and \(x = \gamma\), where \(0 < \alpha < \beta < \gamma\).

1. The Newton-Raphson method is to be used to find the roots of the equation \(\mathrm { f } ( x ) = 0\), with \(x _ { 1 } = k\).
   (a) To which root, if any, would successive approximations converge in each of the cases \(k < 0\) and \(k = 1\) ?
   (b) What happens if \(1 < k < 2\) ?
2. Sketch the curve with equation \(y ^ { 2 } = \mathrm { f } ( x )\). State the coordinates of the points where the curve crosses the \(x\)-axis and the coordinates of any turning points.
3. Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that $$1 + 2 \sinh ^ { 2 } x \equiv \cosh 2 x .$$
4. Solve the equation $$\cosh 2 x - 5 \sinh x = 4$$ giving your answers in logarithmic form.

11 Fig. 11.1 shows the curve with equation \(\mathrm { y } = \mathrm { g } ( \mathrm { x } )\) where \(\mathrm { g } ( x ) = x \sin x + \cos x\) and the curve of the gradient function \(\mathrm { y } = \mathrm { g } ^ { \prime } ( \mathrm { x } )\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\). \begin{figure}[h] \end{figure}

1. Show that the \(x\)-coordinates of the points on the curve \(y = g ( x )\) where the gradient is 1 satisfy the equation \(\frac { 1 } { x } - \cos x = 0\). Fig. 11.2 shows part of the curve with equation \(y = \frac { 1 } { x } - \cos x\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 11.2} \includegraphics[alt={},max width=\textwidth]{60e1e785-c34b-48ef-a63f-13a25fee186e-09_678_1363_424_239}
   \end{figure}
2. Use the Newton-Raphson method with a suitable starting value to find the smallest positive \(x\)-coordinate of a point on the curve \(y = x \sin x + \cos x\) where the gradient is 1 . You should write down at least the following.
   • The iteration you use
   • The starting value
   • The solution correct to \(\mathbf { 4 }\) decimal places
   • Explain why \(x _ { 1 } = 3\) is not a suitable starting value for the Newton-Raphson method in part (b).