question_answer

Let \[f:R\to R\] be a function defined by \[f(x)\,=min\left\{ x+1,\left| x \right|+1 \right\},\] 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 :

\[f\left( x \right)=\min \{x+1,\left| x \right|+1\}\]

\[\Rightarrow f\left( x \right)=x+1\forall x\in \,R\]

Hence, \[f(x)\] is differentiable everywhere for al\[x\in R\].