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

1. A
   $f(x)\ge 1$ for all $x\in \mathbb{R}$
2. B
   $f(x)$ is not differentiable at $x=1$
3. C
   $f(x)$ is differentiable everywhere
4. D
   $f(x)$ is not differentiable at $x=0$