Is the ratio of a decreasing function and an increasing function, a quasi-concave function?

Question

$f(x)$ is a strictly decreasing function and $g(x)$ is a strictly increasing function and positive. Is $h(x) = f(x)/g(x)$ quasi-concave?

Answer 1

Not necessarily -- for example, take $f(x)=-x$ and $g(x)=e^x$.

(In that case the quotient will be quasi-convex, but that also isn't true in general).

If, in addition, $f$ is known to be positive, then $f/g$ is a product of two strictly decreasing positive functions, and is therefore itself strictly decreasing -- and so both quasiconvex and quasiconcave.