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\includegraphics[max width=\textwidth, alt={}, center]{699c5e1e-1476-42cb-b3c4-ca08c4d81cb6-06_485_912_1046_242}
The diagram shows the curve \(M\) with equation \(y = x \mathrm { e } ^ { - 2 x }\).
- Show that \(M\) has a point of inflection at the point \(P\) where \(x = 1\).
The line \(L\) passes through the origin \(O\) and the point \(P\). The shaded region \(R\) is enclosed by the curve \(M\) and the line \(L\).
- Show that the area of \(R\) is given by
\(\frac { 1 } { 4 } \left( a + b \mathrm { e } ^ { - 2 } \right)\),
where \(a\) and \(b\) are integers to be determined.