5 In this question you must show detailed reasoning.
The function f is defined by \(\mathrm { f } ( x ) = \cos x + \sqrt { 3 } \sin x\) with domain \(0 \leqslant x \leqslant 2 \pi\).
- Solve the following equations.
- \(\mathrm { f } ^ { \prime } ( x ) = 0\)
- \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\)
The diagram shows the graph of the gradient function \(y = \mathrm { f } ^ { \prime } ( x )\) for the domain \(0 \leqslant x \leqslant 2 \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{40d40a0b-5b33-4940-b15b-ee03e1291f61-05_583_741_781_242}
- Use your answers to parts (a)(i) and (a)(ii) to find the coordinates of points \(A , B , C\) and \(D\).
- Explain how to use the graph of the gradient function to find the values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
- Using set notation, write down the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing in the domain \(0 \leqslant x \leqslant 2 \pi\).