# If a continuous unimodal function intersects a weakly decreasing function from above, must it be after it reaches its apex?

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 $$f(x). Finally, let $$\hat{x}=\arg \max_{x} f(x)$$. Does it then follow that $$\hat{x}?

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

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

• Yes. Just consider the derivatives. Jul 28 at 1:31