2
$\begingroup$

If we have 2 monotone functions $f$ and $g$ non zero, is it possible that $fg$ has more than one turning point. We can assume wlog that $f$ is increasing and $g$ is decreasing. $\frac{1}{x}e^x$ is an example of one turning point but I can't think of any examples of more than one turning point. Does one exist?

Also if it does, what if we enforce a linear bound on the increasing function?

Thanks.

|cite|improve this question
$\endgroup$
3
$\begingroup$

Let $x\gt 0$. A useful function for constructing examples is the monotone function $f(x)=x+\sin x$. Then $xf(x)$ has infinitely many turning points, as does $\frac{f(x)}{x}$.

|cite|improve this answer
$\endgroup$
  • $\begingroup$ Nice example! Thanks. $\endgroup$ – David Mar 21 '13 at 16:49
  • 1
    $\begingroup$ I think the accepted answer is not appropriate, since $x f(x)$ is a monotonically increasing function. However, its derivative does has many turning points. $\endgroup$ – Mafen Apr 6 '14 at 3:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.