I have a function $f(x)=g(x)h(x)$. I have either of the following two conditions to hold
$g(x)$ has positive value and it is increasing and $h(x)$ has negative values but is decreasing.
$g(x)$ has positive value and it is increasing and $h(x)$ has negative values but is increasing.
Can I say that $f(x)$ is a quasiconcave function? I think I can say that because once the derivative of $f(x)$ becomes negative then it cannot become positive for either of the above two conditions which means that the upper level set will be convex and hence $f(x)$ is quasiconvex. Please let me know if my conclusion is right or wrong.