Under which conditions maximizing a ratio of functions is equivalent to minimizing its reciprocal?

Question

Assume two non-linear functions, $f(x)$, $g(x)$ respectively, both positive and monotone non-decreasing, $f(x)$ is concave, $g(x)$ is convex.

I am trying to maximize their ratio, $\frac{f(x)}{g(x)}$, subject to some inequality constraints. I do not have these functions in closed form but I noticed experimentally that minimizing their reciprocal ratio gives me the same solution as maximizing their ratio. I would like to understand better why this happen. Are there any known conditions for this result?

Answer 1

One sufficient condition: the constraints don't bind and the first ratio is strictly concave and second strictly convex (you can twice derivate these ratios to characterize these). Moreover, the $f$ and $g$ are twice differentiable. Then both the unique maximum of $\frac{f(x)}{g(x)}$ and unique minimum of $\frac{g(x)}{f(x)}$ is given by the first order condition

\begin{equation} f_x(x)g(x)=g_x(x)f(x). \end{equation}