Let $f:R\to R$ be a function defined by $f\left(x\right)=min\{x+1,|x|+1\}$ Then which of the following is true?

$f\left(x\right)\ge 1\text{ for all }x\in R\text{. }$

$f\left(x\right)$ is not differentiable at $x=1$

$f\left(x\right)$ is not differentiable at$x=0$

$f\left(x\right)$ is differentiable everywhere.