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 between - 2 and - 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.