7.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
  \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-3_771_972_1322_557}
\end{center}
\end{figure}

Fig. 2 shows part of the curve $C$ with equation $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x$.\\
The curve crosses the $x$-axis at the origin $O$ and at the points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Factorise $\mathrm { f } ( x )$ completely.
\item Write down the $x$-coordinates of the points $A$ and $B$.
\item Find the gradient of $C$ at $A$.

The region $R$ is bounded by $C$ and the line $O A$, and the region $S$ is bounded by $C$ and the line $A B$.
\item Use integration to find the area of the combined regions $R$ and $S$, shown shaded in Fig.2.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
  \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-4_736_727_338_402}
\end{center}
\end{figure}

Fig. 3 shows a sketch of part of the curve $C$ with equation $y = x ^ { 3 } - 7 x ^ { 2 } + 15 x + 3 , x \geq 0$. The point $P$, on $C$, has $x$-coordinate 1 and the point $Q$ is the minimum turning point of $C$.\\
(a) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(b) Find the coordinates of $Q$.\\
(c) Show that $P Q$ is parallel to the $x$-axis.\\
(d) Calculate the area, shown shaded in Fig. 3, bounded by $C$ and the line $P Q$.
\end{enumerate}

\hfill \mbox{\textit{Edexcel C2  Q7 [17]}}