Is the product of strictly quasiconcave functions quasiconcave? 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.
 A: 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.
A: The product of two positive CONCAVE or log-concave functions is quasi-concave. (Concave implies log-concave, and it is known that the product of log-concave functions is log-concave, hence quasi-concave). 
But the product of two positive quasi-concave functions need not be quasi-concave. Here's a simpler example: $f(x)=x^2$ and $g(x)=9-14x+6x^2$, on the interval $[0,7/6]$. $f$ is increasing, $g$ is decreasing, both positive, but the product has a local max at $0.75$ and a local min at $1$. 
