Related to the concavity of the product of two functions

Question

I have two functions $f(x)$ and $g(x)$. One of them is increasing $f(x)=\frac{1}{a+bx^{-m}}$ and the other is decreasing $g(x)=\frac{1}{c+dx^{m}}$ (where $m<1$). The product $h(x)=f(x)g(x)$ has following derivative $$-\frac{2x^{m-1}(adx^{2m}-bc)}{m(ax^{m}+2)^2(c+dx^{m})^2}$$ where $a>0,b>0,c>0,d>0$. From the derivative we can see that for a small range of $x>0$ the derivative will remain positive while it will become negative for sufficiently high values of $x$ and will never become positive again. So my question is "can we consider $h(x)$ to be concave for values of $x>0$?" Please note that the double derivative of $h(x)$ may not be negative.

Answer 1

A twice differentiable function of one variable is concave on an interval if and only if its second derivative is non-positive. Therefore if $h''(x)$ is positive for any $x > 0$ then $h(x)$ is not concave on $x > 0$.