Question

Statement $1:$ Rolle's theorem is not applicable to $f\left(x\right)=(x-1)|x-1|$ on $[1,2].$
Statement $2:|x-1|$ is not differentiable at $x=1$.

Answer 1

Given : $f\left(x\right)=(x-1)|x-1|$
$\left(1\right):$ $f\left(x\right)$ is continuous on $[1,2].$
$\left(2\right):$ $|x-1|$ is non-differentiable at $x=1$ and $(x-1)$ is differentiable $\forall x\in R.$
$\therefore f\left(x\right)$ is differentiable on $(1,2).$
$\left(3\right):f\left(1\right)=0$ and $f\left(2\right)=1$
Since $f\left(1\right)\ne f\left(2\right)$
$\therefore$ Rolle's theorem is not applicable on $[1,2]$