# Conditions for the product of two monotone functions to be quasiconcave

I have a function $$f(x)=g(x)h(x)$$. I have either of the following two conditions to hold

1. $$g(x)$$ has positive value and it is increasing and $$h(x)$$ has negative values but is decreasing.

2. $$g(x)$$ 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.

Case 1: both $g$ and $-h$ are increasing and positive, therefore their product is increasing. Hence $gh$ is decreasing.
Case 2: both $g$ and $-h$ are decreasing and positive, therefore their product is decreasing. Hence $gh$ is increasing.
In either case $gh$ is monotone, hence quasiconcave.