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\includegraphics[max width=\textwidth, alt={}, center]{918c616a-a0c7-4779-8d3c-84ddf1fa36d6-05_444_757_548_251} The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 3 }\) and \(\mathrm { g } ( x ) = \mathrm { e } ^ { - 2 x }\). The diagram shows the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\). The equation \(\mathrm { f } ( x ) = \mathrm { g } ( x )\) has exactly one real root \(\alpha\).

1. Show that \(\alpha\) satisfies the equation \(\mathrm { h } ( x ) = 0\), where \(\mathrm { h } ( x ) = x ^ { 2 } + 3 - x \mathrm { e } ^ { 2 x }\).
2. Hence show that a Newton-Raphson iterative formula for finding \(\alpha\) can be written in the form $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } \left( 1 - 2 \mathrm { e } ^ { 2 x _ { n } } \right) - 3 } { 2 x _ { n } - \left( 1 + 2 x _ { n } \right) \mathrm { e } ^ { 2 x _ { n } } } .$$
3. Use this iterative formula, with a suitable initial value, to find \(\alpha\) correct to 3 decimal places. Show the result of each iteration. \section*{(d) In this question you must show detailed reasoning.} Find the exact value of \(x\) for which \(\operatorname { fg } ( x ) = \frac { 2 } { 13 }\).