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\begin{figure}[h]
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  \includegraphics[alt={},max width=\textwidth]{2437cecc-f084-4e49-ab36-1c132ba13267-1_480_1058_364_578}
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\caption{Fig. 8}
\end{center}
\end{figure}

Fig. 8 shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = \frac { 1 } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } + 2 }$.\\
(i) Show algebraically that $\mathrm { f } ( x )$ is an even function, and state how this property relates to the curve $y = \mathrm { f } ( x )$.\\
(ii) Find $\mathrm { f } ^ { \prime } ( x )$.\\
(iii) Show that $\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }$.\\
(iv) Hence, using the substitution $u = \mathrm { e } ^ { x } + 1$, or otherwise, find the exact area enclosed by the curve $y = \mathrm { f } ( x )$, the $x$-axis, and the lines $x = 0$ and $x = 1$.\\
(v) Show that there is only one point of intersection of the curves $y = \mathrm { f } ( x )$ and $y = \frac { 1 } { 4 } \mathrm { e } ^ { x }$, and find its coordinates.

\hfill \mbox{\textit{OCR MEI C3  Q1 [18]}}