Question

The points where the function $f\left(x\right)=\left[x\right]+|1-x|,-1\le x\le 3$, where $[.]$ denotes the greatest integer function, is not differentiable are

• A. $x=-1,0,1,2,3$
• B. $x=-1,0,2$
• C. $x=0,1,2,3$
• D. $x=-1,0,1,2$

Answer 1

Answer: (C)

$x=0,1,2,3$

We have,

$f\left(x\right)=\left[x\right]+|1-x|,-1\le x\le 3$

$=\{\begin{array}{c}\begin{array}{cc}-x,& -1\le x<1\end{array}\\ \begin{array}{cc}1-x,& 0\le x<1\end{array}\\ \begin{array}{cc}x,& 1\le x\le 2\end{array}\\ \begin{array}{cc}1+x,& 2\le x<3\end{array}\\ \begin{array}{cc}5,& x=3\end{array}\end{array}$

The only doubtful points are $x=-1,0,1,2$ and $3.$

It can be easily seen that $f\left(x\right)$ is differentiable at $x=-1$ but not differentiable at $x=0,1,2$ and $3.$

Hence, the required points are $0,1,2$ and $3.$