\begin{enumerate}
  \item (a) The function $\mathrm { f } ( x )$ has $\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { u } ( x ) } { \mathrm { v } ( x ) }$. Given that $\mathrm { f } ^ { \prime } ( k ) = 0$, show that $\mathrm { f } ^ { \prime \prime } ( k ) = \frac { \mathrm { u } ^ { \prime } ( k ) } { \mathrm { v } ( k ) }$.
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{dfb57dc0-5831-4bbb-b1e5-58c4798215cb-3_874_879_486_593}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}

(b) The curve $C$ with equation

$$y = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 1 }$$

crosses the $y$-axis at the point $A$. Figure 1 shows a sketch of $C$ together with its 3 asymptotes.\\
(i) Find the coordinates of the point $A$.\\
(ii) Find the equations of the asymptotes of $C$.

The point $P ( a , b ) , a > 0$ and $b > 0$, lies on $C$. The point $Q$ also lies on $C$ with $P Q$ parallel to the $x$-axis and $A P = A Q$.\\
(iii) Show that the area of triangle $P A Q$ is given by $\frac { 5 a ^ { 3 } } { a ^ { 2 } - 1 }$.\\
(iv) Find, as $a$ varies, the minimum area of triangle $P A Q$, giving your answer in its simplest form.

\hfill \mbox{\textit{Edexcel AEA 2009 Q4 [14]}}