Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, unimodal function$^{1}$ and $g:\mathbb{R}\to\mathbb{R}$ be a continuous, weakly decreasing function. Suppose that $f$ intersects $g$ from above at $z\in\mathbb{R}$. That is, $f(z)=g(z)$ and for some $\varepsilon>0$, $f(x)>g(x) \ \forall x\in(z-\varepsilon)$ and $f(x)<g(x) \ \forall x\in(z+\varepsilon)$. Finally, let $\hat{x}=\arg \max_{x} f(x) $. Does it then follow that $\hat{x}<z$?

My intuition tells me "this has to be true!" However, I have a difficult time showing this formally. Any help would be greatly appreciated.

$\quad$ 1. E.g. $f(x)=\frac{ae^{-bx}}{(1+a e^{-bx})^2}$, where $(a,b)\in(0,\infty)^2$.

  • $\begingroup$ Yes. Just consider the derivatives. $\endgroup$ Jul 28 at 1:31

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