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

Question

$\begin{array}{1 1}(a)\;f(x)\;is\;differentiable\;everywhere\\(b)\;f(x)\;is\;not\;differentiable\;at\;x=0\\(c)\;f(x)\geq 1\;for \;all\;x\in R\\(d)\;f(x)\;is \;not\;differentiable\;at\;x=1\end{array}$