# Can the product of two monotone functions have more than one turning point?

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.

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 $\frac{f(x)}{x}$.
• 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. – Mafen Apr 6 '14 at 3:48