Question

If we apply the mean value theorem to $f\left(x\right)=2\sin x+\sin 2x$ then $c=$

• A. $\pi$
• B. $\pi /4$
• C. $\pi /2$
• D. $\pi /3$

Answer 1

Answer: (A), (D)

The correct options are
A $\pi$
D $\pi /3$

Given,

$f\left(x\right)=2\sin x+\sin 2x$

$f^{\prime }\left(x\right)=2\cos x+2\cos 2x$

now, $f^{\prime }\left(c\right)=2\cos c+2\cos 2c=0$

$\cos c=-\cos 2c$

$\cos c=-2{\cos }^{2}c+1$

$2{\cos }^{2}c+\cos c-1=0$

$2{\cos }^{2}c+2\cos c-\cos c-1=0$

$\cos c=-1,\frac{1}{2}$

so,$c=\pi ,\frac{\pi }{3}$