question_answer

Let \[f\left( x \right)=\left[ x \right]+\left| 1-x \right|,-1\le x<3,\] (here [.] denotes greatest integer function). The number of points, where \[f\left( x \right)\] is non-differentiable is

A) 5

B) 0

C) 2

D) none of these

Correct Answer: D

Solution :

\[y=\left[ x \right]+\left| 1-x \right|\operatorname{for}-1\le x<3\]

(i) For \[-1\le x<1,y=-1+1-x=-x\]

(ii) For \[0\le x<1,\,\,\,\,\,\,y=0+1-x=1-x\]

(iii) For \[1\le x<2\,\,\,\,\,\,y=1+x-1=x\]

(iv) For \[2\le x<3,\,\,\,\,\,\,y=2+x-1=x+1.\]

Given function is discontinuous at

\[x=0,1,2,\]

Given function is not differentiable at

\[x=0,1,2.\]