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#### Please solve rd sharma class 12 chapter 14 Mean Value Theoram exercise multiple choice question 9 maths textbook solution

Option (d)

Hint:

Find the derivative of $f(x)$ and then apply the mean value theorem.

Given:

$f(x)=x(x-2),x\in \left [ 1,2 \right ]$

Solution:

$f(x)=x(x-2)$

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

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

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

Using mean value theorem,

$f^{'}(c)=\frac{f(b)-f(a)}{b-a}$

$\Rightarrow \; \; \; \; \; f^{'}(c)=\frac{0-(-1)}{2-1}$                                                $\left [ \because b=2,a=1 \right ]$

$\Rightarrow \; \; \; \; \; f^{'}(c)=\frac{1}{1}$

$\Rightarrow \; \; \; \; \; 2c-2=1$                                $\Rightarrow \; \; \; \; \; \left [ \because f^{'}(c)=2c-2 \right ]$

$\Rightarrow \; \; \; \; \; 2c=1+2$

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

Hence option (d) is correct.