Is it possible to say that product of positive strictly quasiconcave functions is a quasiconcave function?

Question

I have two functions $f_1(x)$ and $f_2(x)$ which are positive over a certain interval and are strictly quasiconcave. Can I say that their product $f(x)=f_1(x)f_2(x)$ is quasiconcave function? Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

The function $\exp(-x^2)$ is quasiconcave, and so is $f(x) = \exp(\exp(-x^2))$, and so are $f(x-10)$ and $f(x+10)$. Note that the function $\exp(-(x-10)^2)+\exp(-(x+10)^2)$ is bimodal, and so is $g(x)=f(x-10)f(x+10)$. So $g$ is the product of two quasiconcave functions but is itself not quasiconcave.