question_answer

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

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