Question

Assertion :Consider the function, $f\left(x\right)=|x-2|+|x-5|,xϵR$
$f^{\prime }\left(4\right)=0$ Reason: $f$ is continuous in $[2,5]$, differntable in $(2,5)$ and $f\left(2\right)=f\left(5\right)$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

given $f\left(x\right)=(x-2)+(x-5),x\in r$

$f^{\prime }\left(x\right)=\frac{|x-2|}{(x-2)}+\frac{|x-5|}{(x-5)}$

at $x=4,x-2>0,\therefore \frac{|x-2|}{(x-2)}=1$

$x-5<0 ,\dfrac{|x-5|}{(x-5)}=-1$

assertion: $\therefore f^{\prime }\left(4\right)=0$

reason : for$x\in (2,5),x-2\ge$

$\therefore |x-2|=x-2$

$x-5<0$

$|x-5|=x-5$

$\therefore f\left(x\right)=3,x\in [2,5]]$

$\therefore f\left(x\right)$is continuous in$[2,5]$ and differentiable in$(2,5)$

$f\left(2\right)=f\left(5\right)=3$

both assertion and reason are correct but reason can't explain assertion correctly.