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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.

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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.

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