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The diagram shows the curve \(y = ( x - 4 ) \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The curve has a gradient of 3 at the point \(P\).
- Show that the \(x\)-coordinate of \(P\) satisfies the equation
$$x = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x }$$
- Verify that the equation in part (i) has a root between \(x = 3.1\) and \(x = 3.3\).
- Use the iterative formula \(x _ { n + 1 } = 2 + 6 \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } }\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.