Question

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

Answer 1

Answer: (C)

We have, $f(x)=\max .[(1-x),(1+x),2]$ for
$x\in (-\infty ,\infty )$
or $f(x)=<br/><br/>\{\begin{array}{ll}<br/><br/>1+x& x>1\\ <br/><br/>2& -1\le x\le 1\\ <br/><br/>1-x& x<-1<br/><br/>\end{array}$
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,
$f(x)=<br/><br/>\{\begin{array}{ll}<br/><br/>1+x& x>1\\ <br/><br/>2& -1\le x\le 1\\ <br/><br/>1-x& x<-1<br/><br/>\end{array}$

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.