8
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{82825739-6239-4afd-9621-538d35c09f28-3_479_1061_342_541}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{figure}
Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 1 } { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } + 2 }\).
- Show algebraically that \(\mathrm { f } ( x )\) is an even function, and state how this property relates to the curve \(y = \mathrm { f } ( x )\).
- Find \(\mathrm { f } ^ { \prime } ( x )\).
- Show that \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } + 1 \right) ^ { 2 } }\).
- 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 \(x = 1\).
- 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.