question_answer

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

A) f(x) is differentiable everywhere

B) f(x) is not differentiable at \[x=0\]

C) \[f(x)\ge 1\] for all \[x\in R\]

D) f(x) is not differentiable at \[x=1\]

Correct Answer: A

Solution :

\[f(x)=\,\,min\,\left\{ x+1,\,\left| x \right|+1 \right\}\] \[\Rightarrow \,\,\,f(x)=x\,+1\forall x\in R\] Hence, f(x) is differentiable everywhere for all\[\operatorname{x}\in R\].