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The diagram shows the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\) and its maximum point \(M\).
- Find the \(x\)-coordinate of \(M\).
- Show that the tangent to the curve at the point where \(x = 1\) passes through the origin.
- Use the trapezium rule, with two intervals, to estimate the value of
$$\int _ { 1 } ^ { 3 } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$
giving your answer correct to 2 decimal places.