What is an example for a strictly increasing strictly concave differentiable function with no lower or upper bound on $\Bbb R$?

Question

I am looking for an example of a function on $\Bbb R$ that is

"symmetric to the origin" stands for even or for odd? Anyway, the solutions set is empty if you add such a constraint. — Jack D'Aurizio, Oct 10 '16 at 18:00
@JackD'Aurizio I meant actually axis symmetry to the -x axis. — Daniel, Oct 10 '16 at 18:06
Symmetric with respect to the $x$-axis? So $f(x)$ is at the same time $a$ and $-a$ ?! — Jack D'Aurizio, Oct 10 '16 at 18:08
@JackD'Aurizio the $-x$ axis I meant. At least that I can paint on a paper. — Daniel, Oct 10 '16 at 18:12

score 4 · Accepted Answer · answered Oct 10 '16 at 17:51

4

You might try $$f(x)=x-e^{-x} $$

answered Oct 10 '16 at 17:51

Hagen von Eitzen

Thanks. That seems to work. – Daniel Oct 10 '16 at 17:53
Actually, now that I see it. Is it possible to have it also symmetric to the origin? – Daniel Oct 10 '16 at 17:55
1

@Daniel: no odd or even function may fulfill the given constraints. – Jack D'Aurizio Oct 10 '16 at 17:58
@JackD'Aurizio Wow. Interesting that this is actually impossible. – Daniel Oct 10 '16 at 17:59
@Daniel: if you try to draw a graph, that is pretty trivial, actually. – Jack D'Aurizio Oct 10 '16 at 18:01
Oh, I mixed it up – Daniel Oct 10 '16 at 18:02
Symmetrici to the origin means $f(-x)=-f(x)$, hence $f''(-x)=-f''(x)$, so you cannot have strictly concave – Hagen von Eitzen Oct 10 '16 at 18:05
@JackD'Aurizio What I meant was a symmetry at the axis given by -x. Not sure why I thought this has to do with point symmetry. – Daniel Oct 10 '16 at 18:05
@HagenvonEitzen Thanks. I got confused. Have updated it in the comment. – Daniel Oct 10 '16 at 18:07

1 Answers1