8.
\includegraphics[max width=\textwidth, alt={}, center]{a86f277c-a2ec-4ba0-ab08-575cad2a5e53-3_424_698_246_479}
The diagram shows the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\) where
$$\mathrm { f } ( x ) = \frac { \cos x } { 2 - \sin x } , \quad x \in \mathbb { R }$$
- Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - 2 \sin x } { ( 2 - \sin x ) ^ { 2 } }\).
- Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = \pi\).
- Find the minimum and maximum values of \(\mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\).
- Explain why your answers to part (c) are the minimum and maximum values of \(\mathrm { f } ( x )\) for all real values of \(x\).