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?



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 $\frac{f(x)}{x}$.

  • $\begingroup$ Nice example! Thanks. $\endgroup$ – David Mar 21 '13 at 16:49
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    $\begingroup$ 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

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