#### need solution for rd sharma maths class 12 chapter 14 Mean Value Theoram exercise multiple choice question 11

Option (b)

Hint:

You must know about the concept of Rolle’s Theorem.

Given:

$f(x)=e^{x}\sin x,x\in \left [ 0,\pi \right ]$

Solution:

$f(x)=e^{x}\sin x$

$\Rightarrow f^{'}(x)=e^{x}\cos x-\sin x\left ( e^{x} \right )$

$\Rightarrow f^{'}(x)=e^{x}(\cos x -\sin x )$

$\Rightarrow f^{'}(c)=e^{c}(\cos c -\sin c )$

As Rolle’s Theorem,

$\Rightarrow f^{'}(c)=0$

$\Rightarrow e^{c}(\cos c -\sin c )=0$

$\Rightarrow \cos c -\sin c =0$

$\Rightarrow \cos c=\sin c$

$\Rightarrow \sin \left ( \frac{\pi }{2}-c \right )=\sin c$

$\Rightarrow c=\frac{\pi }{2}-c$

$\Rightarrow 2c=\frac{\pi }{2}$

$\Rightarrow c=\frac{\pi }{4}$

Option (b) is correct.