Question

Consider the function, $f\left(x\right)=|x-2|+|x-5|,x\in R$.
Statement-$1:f^{\prime }\left(4\right)=0$
Statement-$2:f$ is continuous in $[2,5]$, differentiable in $(2,5)$ and $f\left(2\right)=f\left(5\right)$.

• A. Statement$-1$ is false, Statement$-2$ is true.
• B. Statement$-1$ is true, Statement$-2$ is true; statement$-2$ is a correct explanation for Statement$-1$.
• C. Statement$-1$ is true, Statement$-2$ is true; statement$-2$ is not a correct explanation for Statement$-1$.
• D. Statement$-1$ is true, Statement$-2$ is false.

Answer 1

The correct option is C Statement$-1$ is true, Statement$-2$ is true; statement$-2$ is not a correct explanation for Statement$-1$.

$f\left(x\right)=3$ when $2\le x\le 5$
$f^{\prime }\left(x\right)=0$ when $2<x<5$ (because $f\left(x\right)$ is a constant function)
$f^{\prime }\left(4\right)=0$
Therefore both statements are true but statement$-2$ is not correct explanation for statement$-1.$