<s>[INST] What is the maximum value of $ f(x)=\frac{2 \sin (3 x)}{3 \sin (x)+3 \sqrt{3} \cos (x)} (\frac{\pi}{3}I would appreciate if somebody could help me with the following problem
Q: What is the maximum value of $$ f(x)=\frac{2 \sin (3 x)}{3 \sin (x)+3 \sqrt{3} \cos (x)} (\frac{\pi}{3}<x<\frac{2\pi}{3})$$
I have done my work here 
$$f'(x)=\frac{2 \left(2 \sqrt{3} \cos (2 x)+\sqrt{3} \cos (4 x)-8 \sin ^3(x) \cos (x)\right)}{3 \left(\sin (x)+\sqrt{3} \cos (x)\right)^2}=0$$
I tried to solve problems but I couldn't make further progress
 [/INST] $$f(x) = \frac{1}{3}\cdot\frac{\sin(3x)}{\sin\left(x+\frac{\pi}{3}\right)} $$
is a negative function on the interval $\left(\frac{\pi}{3},\frac{2\pi}{3}\right)$, while $f\left(\frac{\pi}{3}\right)=0$.
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