Question

The value of $a$ for which the function $f\left(x\right)=a\sin x+\frac{1}{3}\sin 3x$ has an extremum at $x=\frac{\pi }{3}$ is

• A. $1$
• B. $-1$
• C. $0$
• D. $2$

Answer 1

The correct option is D $2$

$f\left(x\right)=a\sin x+\frac{1}{3}\sin 3x$

$\Rightarrow f^{\prime }\left(x\right)=a\cos x+\frac{1}{3}\cos 3x\times 3$

$\Rightarrow f^{\prime }\left(x\right)=a\cos x+\cos 3x$

For extremum at $\frac{\pi }{3}$

$f^{\prime }\left(\frac{\pi }{3}\right)=0$

$\Rightarrow a\cos \left(\frac{\pi }{3}\right)+\cos 3\left(\frac{\pi }{3}\right)=0$

$\Rightarrow \frac{a}{2}-1=0$

$\Rightarrow a=2$