Question

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

• A. $f\left(x\right)$ is differentiable everywhere
• B. $f\left(x\right)$ is not differentiable at $x=0$
• C. $f\left(x\right)\ge 1$ for all $x\in R$
• D. $f\left(x\right)$ is not differentiable at $x=1$

Answer 1

The correct option is A $f\left(x\right)$ is differentiable everywhere

Graphs of $y=x+1$ and $y=|x|+1$ are:-

$\Rightarrow min\{x+1,|x|+1\}=x+1$ every where

$\Rightarrow$ Option (A).