4 Answer the whole of this question on the insert provided.
\includegraphics[max width=\textwidth, alt={}]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-02_887_1273_1137_438}
The sketch shows the curve with equation \(y = \mathrm { F } ( x )\) and the line \(y = x\). The equation \(x = \mathrm { F } ( x )\) has roots \(x = \alpha\) and \(x = \beta\) as shown.
- Use the copy of the sketch on the insert to show how an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\), with starting value \(x _ { 1 }\) such that \(0 < x _ { 1 } < \alpha\) as shown, converges to the root \(x = \alpha\).
- State what happens in the iteration in the following two cases.
(a) \(x _ { 1 }\) is chosen such that \(\alpha < x _ { 1 } < \beta\).
(b) \(x _ { 1 }\) is chosen such that \(x _ { 1 } > \beta\).
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4 -
\includegraphics[max width=\textwidth, alt={}, center]{0ec9c4ff-8622-4dda-a000-6ffe36f38023-03_873_1259_274_484}
- (a) \(\_\_\_\_\)
(b) \(\_\_\_\_\)
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