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) =
\begin{cases}
1+x & x > 1 \\
2 & -1 \le x \le 1 \\
1-x & x < -1
\end{cases} $
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) =
\begin{cases}
1+x & x > 1 \\
2 & -1 \le x \le 1 \\
1-x & x < -1
\end{cases} $

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.