Question

The difference between the greatest and the least value of $f(x)={\cos }^2\frac{x}{2}\sin x,x\in [0,\pi ]$ is

• A. $\frac{3\sqrt{3}}{8}$
• B. $\frac{\sqrt{3}}{8}$
• C. $\frac{3}{8}$
• D. $\frac{1}{2\sqrt{2}}$

Answer 1

Answer: (A)

$f(x)=\frac{\sin x(1+\cos x)}{2};f^{\prime }(x)=\frac{1}{2}(\cos x+\cos 2x)\Rightarrow \cos x+2{\cos }^{2}x-1=0$
$\Rightarrow 2{\cos }^{2}x+\cos x-1=0\Rightarrow 2{\cos }^{2}x+2\cos x-\cos x-1=0$
$\Rightarrow 2\cos x(\cos x+1)-(1+\cos x)=0\Rightarrow \cos x=1/2\text{ or }\cos x=-1$
$\Rightarrow x=\pi \text{ or }\frac{\pi }{3}$