Question

Consider the functions, $f\left(x\right)=|x-2|+|x-5|,x\in R$
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

Answer: (B)

Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1

$f\left(x\right)=7-2x;x<2$
$=3;$ $2\le x\le 5$
$=2x-7;x>5$
$f\left(x\right)$ is a constant function in $[2,5]$,
$f$ is continuous in $[2,5]$ and differentiable in $(2,5)$ and $f\left(2\right)=f\left(5\right)$
$f^{\prime }\left(4\right)=0$ by Rolle's theorem
Hence, option 'B' is correct.