8. A curve has the equation \(y = ( 2 x + 3 ) \mathrm { e } ^ { - x }\).
- Find the exact coordinates of the stationary point of the curve.
The curve crosses the \(y\)-axis at the point \(P\).
- Find an equation for the normal to the curve at \(P\).
The normal to the curve at \(P\) meets the curve again at \(Q\).
- Show that the \(x\)-coordinate of \(Q\) lies in the interval \([ - 2 , - 1 ]\).
- Use the iterative formula
$$x _ { n + 1 } = \frac { 3 - 3 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { x _ { n } } - 2 }$$
with \(x _ { 0 } = - 1\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 2 decimal places.
- Show that your value for \(x _ { 4 }\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places.
END