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\includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-3_655_685_262_730}
The diagram shows the curve \(y = x \mathrm { e } ^ { 2 x }\) and its minimum point \(M\).
- Find the exact coordinates of \(M\).
- Show that the curve intersects the line \(y = 20\) at the point whose \(x\)-coordinate is the root of the equation
$$x = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x } \right)$$
- Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \ln \left( \frac { 20 } { x _ { n } } \right)$$
with initial value \(x _ { 1 } = 1.3\), to calculate the root correct to 2 decimal places, giving the result of each iteration to 4 decimal places.