Question

Let $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{cc}max\{-x,x+2\},& x<0\\ 2,& 0\le x<1\\ 3-x,& x\ge 1\end{array}.$ Then

• A. $f\left(x\right)$ is continuous for all $x\in R$
• B. $f^{\prime }\left(x\right)>0$ for $-1<x<0$
• C. $f\left(x\right)$ is non-differentiable at $x=0$
• D. $f\left(x\right)$ is discontinuous at $x=0$

Answer 1

Answer: (A), (B), (C)

$f\left(x\right)$ is non-differentiable at $x=0$

$f\left(x\right)=\{\begin{array}{cc}-x& x\le -1\\ x+2,& -1<x<0\\ 2,& 0\le x<1\\ 3-x,& x\ge 1\end{array}$

$f^{\prime }\left(x\right)=\{\begin{array}{cc}-1& x\le -1\\ 1,& -1<x<0\\ 0,& 0\le x<1\\ -1,& x\ge 1\end{array}$

$f^{\prime }\left(x\right)=1>0$ for $-1<x<0$

LHD $\ne$ RHD at $x=0$
Hence, $f\left(x\right)$ is non-differentiable at $x=0$