There are two functions, $f(x)$ and $g(x)$ defined on the interval $[a,b]$ and I am interested in their product, $$h(x)=f(x)g(x)$$ $f(x)$ and $g(x)$ are both positive, $f'(x)>0$ and $g'(x)<0$. I would like to find further assumptions needed to ensure $h(x)$ is concave and has a unique maximum on the interval.
My approach so far has been to look at what we need for the maximum to exist, i.e. $h'(x)=0$ for some $x\in(a,b)$ and $h''(x)<0$. This gives me two conditions: $$g(x)f'(x)+f(x)g'(x)=0$$ for some $x\in(a,b)$ and $$g(x)f''(x)+2f'(x)g'(x)+f(x)g''(x)<0$$ The second condition can easily be satisfied by the assumption that $f(x)$ and $g(x)$ are both concave functions, which I am happy to make. How about the first condition? How can this be satisfied other than by making an explicit assumption such as $\frac{g(x)}{g'(x)}+\frac{f(x)}{f'(x)}=0$ for some $x\in(a,b)$? And what does such an assumption really mean (how to interpret it)?
A concave differentiable function $h$ on the interval $[a,b]$ has a maximum in $(a,b)$ if $h'(a) > 0$ and $h'(b) < 0$. The maximum is unique if there is no interval of positive length on which $h$ is constant. For that it suffices that $h'' < 0$.