$ = 2[3\cos x - \sin x] > 0$ in $\left[ {\frac{{5\pi }}{3},\frac{{7\pi }}{4}} \right]$
$\therefore$ $f(x)$ is an increasing function, hence $f(x)$ has greatest value at $x = \frac{{7\pi }}{4}$
$\therefore$ Greatest $f(x) = f\left( {\frac{{7\pi }}{4}} \right) = [6\sin t + 2\cos t]_{5\pi /3}^{7\pi /4}$
$ = - 6\frac{1}{{\sqrt 2 }} + \frac{2}{{\sqrt 2 }} + 6\frac{{\sqrt 3 }}{2} - 1 $
$= 3\sqrt 3 - 2\sqrt 2 - 1$.
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