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

Question

( A ) $f(x) \geq 1$ for all $x \in R$
( B ) $f(x)$ is not differentiable at $x= 0$
( C ) $f(x)$ is differentiable everywhere
( D ) $f(x)$ is not differentiable at $x=1$