Checking differentiability

Question

I have a question to share.

Find the points at which the following function is not differentiable: $$f(x)=\max \lbrace 1-x,1+x,2\rbrace \quad\forall x\in \mathbb{R}$$

I have joined this site recently and do not have enough reputation. So I am answering it here itself:

Observing the graphs of $y=1-x,y=1+x$ and $y=2$ we see that the above function is same as the following:

$$f(x)=\left\lbrace \begin{array}{cl} 1-x, & x<-1 \\ 2, & -1\leq x\leq 1\\ 1+x, & x>1 \end{array}\right.$$

and then it can be shown that $f$ is not differentiable at $x=-1$ and $x=1$.

So what is your goal here? Do you want us to help you by answering the question or just verifying it? — 5xum, Jun 11 '14 at 10:56
Just wanted to share it as I found it interesting. You may verify it too and put comments if any. — Debashish, Jun 11 '14 at 11:01
Small hint: instead of using commas in your "cases" block, use "&", to get nicely formatted "conditions." — John Hughes, Jun 11 '14 at 11:03
I have "voted up" your question so you should have enough reputation now to convert the latter part of your question into an answer if you so choose. — apnorton, Jun 11 '14 at 11:03
@ anorton ... wow ! thanks a lot but still I have only 13 reputation..I think I need 2 more to do that. Hoping to get some more soon. — Debashish, Jun 11 '14 at 11:05
@ anorton ... Oh I could post my answer now. Thanks for your "up vote" — Debashish, Jun 11 '14 at 11:15

score 0 · Accepted Answer · answered Jun 11 '14 at 11:14

I can post the answer now:

Observing the graphs of $y=1-x,y=1+x$ and $y=2$ we see that the above function is same as the following:

$$f(x)=\left\lbrace \begin{array}{cl} 1-x, & x<-1 \\ 2, & -1\leq x\leq 1\\ 1+x, & x>1 \end{array}\right.$$

and then it can be shown that $f$ is not differentiable at $x=-1$ and $x=1$.

1 Answers1