I know that the following function might have zero local extrema (when it is strictly increasing), 1 local extremum (when it is increasing and then decreasing) or 2 local extrema (when it is increasing and then decreasing and then increasing again). I want to know if it is possible to show and prove which of these cases happen depending on the values of the coefficients $a,\dotsc,e$, where $a,b,c,d$ are positive real numbers and $e$ is a real number:
$$f(x)=\dfrac{a\log(1+bx)}{cx+d}+ e\log(1+bx), \quad x\geq0$$
I know that the above question is equivalent to showing if $$g(x)=\dfrac{df(x)}{dx}$$ has zero, one or two roots. So, whichever is easier to show, will be fine. Please let me know if you have any suggestions.
Thanks in advance.