sum of monotonic increasing and monotonic decreasing functions I have a question regarding sum of monotinic increasing and decreasing functions. Would appreciate very much any help/direction:
Consider an interval $x \in [x_0,x_1]$. Assume there are two functions $f(x)$ and $g(x)$ with $f'(x)\geq 0$ and $g'(x)\leq 0$. We know that $f(x_0)\leq 0$, $f(x_1)\geq 0$, but $g(x)\geq 0$ for all $x \in [x_0,x_1]$. I want to show that $q(x) \equiv f(x)+g(x)$ will cross zero only once. We know that $q(x_0)\leq 0$ and $q(x_1)\geq 0$.
Is there a ready result that shows it or how to proceed to show that? Many thanks!!!

Alas, the answer is no.
$$f(x)=\begin{cases}-4& x\in[0,2]\\ -2& x\in [2,4]\\0& x\in[4,6]\end{cases}$$
$$g(x)=\begin{cases}5 & x\in [0,1]\\3& x\in[1,3]\\1& x\in[3,5]\\ 0 & x\in[5,6]\end{cases}$$
$$q(x)=\begin{cases} 1 & x\in [0,1]\\ -1& x\in[1,2]\\ 1 & x\in[2,3]\\ -1 & x\in[3,4]\\ 1& x\in[4,5]\\ 0 & x\in[5,6]\end{cases}$$
This example could be made continuous and strictly monotone with some tweaking.
