5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{6e307391-198f-4ea9-99ed-6ef184fca0f7-5_846_693_246_612}
\end{figure}
The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). Figure 1 shows the part of \(C\) for which \(0 \leq x \leq 2\).
Given that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } - 2 x ^ { 2 } ,$$
and that \(C\) has a single maximum, at \(x = k\),
- show that \(1.48 < k < 1.49\).
Given also that the point \(( 0,5 )\) lies on \(C\),
- find \(\mathrm { f } ( x )\).
The finite region \(R\) is bounded by \(C\), the coordinate axes and the line \(x = 2\).
- Use integration to find the exact area of \(R\).
(4)