Definition 1: Monotonicity
A function $\displaystyle g( \cdot )$ is monotonically increasing if for all pairs of inputs $\displaystyle x_{1}$ and $\displaystyle x_{2}$, such that $\displaystyle x_{1} < x_{2}$, $\displaystyle g( x_{1}) < g( x_{2})$. Similarly, $\displaystyle g$ is monotonically decreasing if for all pairs of inputs $\displaystyle x_{1}$ and $\displaystyle x_{2}$, such that $\displaystyle x_{1} < x_{2}$, $\displaystyle g( x_{1}) >g( x_{2})$. These conditions define strict monotonicity. A weaker version of monotonicity replaces ineqality symbols $\displaystyle < $ and $\displaystyle >$with $\displaystyle \leqslant $and $\displaystyle \geqslant $ so that $\displaystyle g( x_{1}) \leqslant g( x_{2})$ and $\displaystyle g( x_{1}) \geqslant g( x_{2})$, respectively, for all $\displaystyle x_{1} < x_{2}$.
Definition 2: Inverted-U
The inverted-$\displaystyle U$ relationship can be defined as unimodality with a maximum. Namely, $\displaystyle f:[ x_{min} ,x_{max}] \ \rightarrow \mathbb{R}$ has an inverted-$\displaystyle U$ relationship with $\displaystyle x$, if there exists some value $\displaystyle x_{*}$ such that $\displaystyle f( x)$ is monotonically increasing in the interval $\displaystyle [ x_{min} ,x_{*}]$, and monotonically decreasing on $\displaystyle [ x_{*} ,x_{max}]$. Then $\displaystyle f( x_{*})$ is the maximum in $\displaystyle [ x_{min} ,x_{max}]$. I also have an intuition that if $g_+$ and $g_-$ are not strictly monotonic, then there could be an interval $[y_*, y_{**}] \in [x_{min} ,x_{*}]$ where $f(y)$ would be a kind of "maximum plateau".
Statement
Function $\displaystyle f( x) =g_{+}( x) \cdot g_{-}( x)$ has an inverted-U shape if all of the following conditions are true:
- Monotonic and antagonistic factors. $\displaystyle g_{+}$ is a monotonically increasing function of $\displaystyle x$, $\displaystyle g_{-}$ is a monotonically decreasing function of $\displaystyle x$.
- Shared bounded domain. Functions $\displaystyle f$, $\displaystyle g_{+}$, and $\displaystyle g_{-}$ have a shared bounded non-negative real domain, i.e., $\displaystyle [ x_{min} ,x_{max}] \in \mathbb{R}_{\geqslant 0}$.
- Finite non-negative ranges. $\displaystyle f$, $\displaystyle g_{+}$, $\displaystyle g_{-} :[ x_{min} ,x_{max}] \ \rightarrow \mathbb{R}_{\geqslant 0}$.
- Zero minima. Let $\displaystyle g_{+}( x_{min}) =g_{-}( x_{max}) =0$.
How do I prove my statement (or what conditions should I add/change to make it true)?
Some disclaimers. I have practically no experience with proofs and limited mathematical education. Apologies in advance if the question is too wordy/imprecise/nonsensical. I am also not asking to write the proof for me, although it is welcome. Given my limited experience, it would be helpful to get at least some sense of direction.