A curve has equation \(y = \mathrm { e } ^ { 2 x } - 10 \mathrm { e } ^ { x } + 12 x\).
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(2 marks)
Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
(1 mark)
The points \(P\) and \(Q\) are the stationary points of the curve.
Show that the \(x\)-coordinates of \(P\) and \(Q\) are given by the solutions of the equation
$$\mathrm { e } ^ { 2 x } - 5 \mathrm { e } ^ { x } + 6 = 0$$
(1 mark)
By using the substitution \(z = \mathrm { e } ^ { x }\), or otherwise, show that the \(x\)-coordinates of \(P\) and \(Q\) are \(\ln 2\) and \(\ln 3\).
Find the \(y\)-coordinates of \(P\) and \(Q\), giving each of your answers in the form \(m + 12 \ln n\), where \(m\) and \(n\) are integers.
Using the answer to part (a)(ii), determine the nature of each stationary point.