If the function $f\left(x\right) = a\sin \left(x\right) + \frac{1}{3}\sin \left(3x\right)$ attains maximum value at $x=\frac{\pi }{3}$ then ‘a’ equals

$\begin{array}{rcl}\mathrm{f}\left(\mathrm{x}\right)& =& \mathrm{asinx}+\frac{1}{3}\sin 3\mathrm{x}\\ & \Rightarrow & {\mathrm{f}}^{\mathrm{I}}\left(\mathrm{x}\right)=\mathrm{acosx}+\frac{1}{3}\left(\cos 3\mathrm{x}\right)3\\ & =& \mathrm{acosx}+\cos 3\mathrm{x}\end{array}$

Given $\mathrm{f}\left(\mathrm{x}\right)$ is maximum at $\mathrm{x}=\mathrm{\pi }/3$

$$\begin{gathered} \Rightarrow {\mathrm{f}}^{\mathrm{I}}\left(\mathrm{\pi }/3\right)=0 \\ \Rightarrow \mathrm{a}\left(1/2\right)+\left(-1\right)=0 \\ \Rightarrow \mathrm{a}/2=1\Rightarrow \mathrm{a}=2 \end{gathered}$$