Is the product of strictly quasiconcave functions quasiconcave?

Question

I have two functions $f_1(x)$ and $f_2(x)$ which are strictly quasiconcave (since they have positive derivative before a certain point and negative derivative after that point. The points where derivative become zero can be different for each of the functions). I want to ask whether $f(x)=f_1(x)f_2(x)$ will be a quasiconcave function or not? Any help in this regard will be highly appreciated. Thanks in advance.

Answer 1

The answer is no. A simple example will suffice. Consider the L 1/2 norm and a shifted L 1/2 norm.

Now, if two non-negative quasi-convex functions have the same optimal point, then the product is quasi-convex. I think there are some other cases where this works.

Another question might be: is it possible to combine two quasi-convex functions into a new quasi-convex function. And the answer is, yes it is! Suppose that $f_1$ and $f_2$ are quasi-convex. Then $f=\max(f_1,f_2)$ is a quasi-convex function.