The maximum value of 5 $\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+3$ is

We have,

$\begin{array}{l}5\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+3\\ =5\cos \theta +3\cos \theta \cos \frac{\pi }{3}-3\sin \theta \sin \frac{\pi }{3}+3\\ =\frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3\end{array}$

Now,

$\begin{array}{l}-\sqrt{{\left(\frac{13}{2}\right)}^2}+{\left(\frac{3\sqrt{3}}{2}\right)}^2\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le \sqrt{{\left(\frac{13}{2}\right)}^2+{\left(\frac{3\sqrt{3}}{2}\right)}^2}\\ \end{array}$

$\begin{array}{l}\Rightarrow -7\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta \le 7\text{ for all }\theta \\ \Rightarrow -4\le \frac{13}{2}\cos \theta -\frac{3\sqrt{3}}{2}\sin \theta +3\le 10\text{ for all }\theta \\ \Rightarrow -4\le 5\cos \theta +3\cos \left(\theta +\frac{\pi }{3}\right)+3\le 10\text{ for all }\theta \end{array}$