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\includegraphics[max width=\textwidth, alt={}, center]{72d50061-ead5-466a-96fc-2203438d1407-3_296_675_945_735} The diagram shows part of the curve \(y = \frac { x ^ { 2 } } { 1 + \mathrm { e } ^ { 3 x } }\) and its maximum point \(M\). The \(x\)-coordinate of \(M\) is denoted by \(m\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m\) satisfies the equation \(x = \frac { 2 } { 3 } \left( 1 + \mathrm { e } ^ { - 3 x } \right)\).
2. Show by calculation that \(m\) lies between 0.7 and 0.8 .
3. Use an iterative formula based on the equation in part (i) to find \(m\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.