question_answer

The function \[F(x)=\max [(1-x),(1+x),2],x\in (-\infty ,\infty )\]is:

A)  continuous at all points

B)                   differentiable at all points

C)                   differentiable at all points except at \[x=1\] and \[x=-1\]

D)                   none of the above

Correct Answer: C

Solution :

We have, \[f(x)=\max .[(1-x),(1+x),2]\]for \[x\in (-\infty ,\infty )\] or            \[fx=\left\{ \begin{matrix}    1+x, & x>1  \\    2, & -1\le x\le 1  \\    1-x, & x<-1  \\ \end{matrix} \right.\] Since,\[f(x)\] is a polynomial and constant function which is defined for every values of\[x,\] therefore \[f(x)\]is continuous for all values of\[x\] \[\therefore \]\[f(x)\] is differentiable for all values of \[x\] except at \[x=1\]and \[-1.\] Alternate Solution: We have, \[f(x)=\left\{ \begin{matrix}    1+x, & x>1  \\    2, & -1\le x\le 1  \\    1-x, & x<-1  \\ \end{matrix} \right.\] It is clear from the figure that \[f(x)\] is continuous everywhere and \[f(x)\] is differentiable everywhere except at \[x=1,-1.\] Note: Every differentiable function is continuous but   every   continuous   function   is   not differentiable.