Question

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

• A. $\frac{9}{2}$
• B. $\frac{5}{2}$
• C. $\frac{3\sqrt{3}}{2}$
• D. $\frac{3}{2}$

Answer 1

The correct option is C $\frac{3\sqrt{3}}{2}$

$f\left(x\right)=2\sin x+\sin 2x$
Differentiating, we get $f^{\prime }\left(x\right)=2\cos x+2\cos 2x$
Equating it to 0, we have $\cos x=1-2{\cos }^{2}x$
$\Rightarrow \cos x-1+2{\cos }^{2}x=0$
$\Rightarrow \cos x=\frac{-1\pm \sqrt{1+8}}{4}$
$\Rightarrow \cos x=-1$ or $0.5$
Thus, $\theta$ is either $\pi$ or $\frac{\pi }{3}$
So, the value of expression becomes $0$ or $2\times \frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}=3\frac{\sqrt{3}}{2}$