$\begin{aligned} & \text { } \because f(x)=a \sin x+\frac{1}{3} \sin 3 x \\ & f(x)=a \cos x+\frac{1}{3} \times 3 \cos 3 x \\ & \Rightarrow f(x)=a \cos x+\cos 3 x \\ & \because f(x) \text { has an extremum at } x=\frac{\pi}{3} \\ & \therefore a x \cos \left(\frac{\pi}{3}\right)+\cos (\pi)=0 \\ & \frac{a}{2}-1=0 \Rightarrow \frac{a}{2}=1 \Rightarrow a=2\end{aligned}$