8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-11_1112_1211_280_386}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C\) with the equation \(y = \mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 9 x + 9 \right) e ^ { - x } , \quad x \in R$$
The curve has a minimum turning point at \(A\) and a maximum turning point at \(B\) as shown in the figure above.
a. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
b. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \left( 2 x ^ { 2 } - 13 x + 18 \right) e ^ { - x }\)
c. Hence find the exact coordinates of the turning points of \(C\).
The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation
$$y = a \mathrm { f } ( x ) + b , \quad x \geq 0$$
The range of the graph with equation \(y = a \mathrm { f } ( x ) + b\) is \(0 \leq y \leq 9 e ^ { 2 } + 1\)
Given that \(a\) and \(b\) are constants.
d. find the value of \(a\) and the value of \(b\).