Question

The greatest value of the function $f\left(x\right)=2\sin x+\sin 2x$ in the interval $[0,\frac{3\pi }{2}]$ is

• A. $\frac{3\sqrt{3}}{2}$
• B. $3$
• C. $\frac{3}{2}$
• D. None of the above

Answer 1

Answer: (A)

$\frac{3\sqrt{3}}{2}$

$f\left(x\right)=2\sin x+\sin 2x$
$f^{\prime }\left(x\right)=2\cos x+2\cos 2x$
For maxima or minima, $f^{\prime }\left(x\right)=0$
$\cos x+\cos 2x=0$
$\cos \frac{3x}{2}\cos \frac{x}{2}=0$
$\Rightarrow x=\frac{\pi }{3},\pi$
Now, $f\left(0\right)=0,f\left(\frac{\pi }{3}\right)=\frac{3\sqrt{3}}{2},f\left(\pi \right)=0,f\left(\frac{3\pi }{2}\right)=-2$
Hence, greatest value of $f\left(x\right)$ is $\frac{3\sqrt{3}}{2}$