#### provide solution for rd sharma maths class 12 chapter 14 Mean Value Theoram exercise  multiple choice question 10

Option (a)

Hint:

Differentiate the given function and then apply Rolle’s Theorem.

Given:

$f(x)=x^{3}-3x,x\in \left [ 0,\sqrt{3} \right ]$

Solution:

$f(x)=x^{3}-3x,x\in \left [ 0,\sqrt{3} \right ]$

$f(x)=x^{3}-3x$

$\Rightarrow \; \; \; \; \; f^{'}(x)=3x^{2}-3$

$\Rightarrow \; \; \; \; \; f^{'}(c)=3c^{2}-3$

Applying Rolle’s Theorem,

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

$\Rightarrow \; \; \; \; \; 3c^{2}-3=0$

$\Rightarrow \; \; \; \; \; 3c^{2}=3$

$\Rightarrow \; \; \; \; \; c^{2}=1$

$\Rightarrow \; \; \; \; \; c=\pm 1$

$\Rightarrow \; \; \; \; \; c=1$                                   $\left [ \because c\in \left [ 0,\sqrt{3} \right ] \right ]$

Hence option (a) is correct.