7.
$$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$
- Show that \(\mathrm { f } ( x )\) may be written in the form
$$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
- Find the range of the function \(\mathrm { f } ( x )\).
The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure} - Find the coordinates of all the maximum and minimum points on this curve.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The region \(R\), bounded by \(y = 2\) and \(y = \mathrm { f } ( x )\), is shown shaded in Figure 3. - Find the area of \(R\).