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

Question

$\begin {array} {1 1} (A)\;f(x)\geq 1\: for \: all \: x\: \in \: R & \quad (B)\; f(x)\: is\: not\: differentiable\: at\: x = 1 \\ (C)\; f(x)\: is \: differentiable \: everywhere & \quad (D)\;f(x)\: is \: not \: differentiable \: at\: x=0 \end {array}$