Awainaideannagi

2022-07-16

Sum of monotonic increasing and monotonic decreasing functions
Consider an interval $x\in \left[{x}_{0},{x}_{1}\right]$. Assume there are two functions f(x) and g(x) with ${f}^{\prime }\left(x\right)\ge 0$ and ${g}^{\prime }\left(x\right)\le 0$. We know that $f\left({x}_{0}\right)\le 0$, $f\left({x}_{1}\right)\ge 0$, but $g\left(x\right)\ge 0$ for all $x\in \left[{x}_{0},{x}_{1}\right]$. I want to show that $q\left(x\right)\equiv f\left(x\right)+g\left(x\right)$ will cross zero only once. We know that $q\left({x}_{0}\right)\le 0$ and $q\left({x}_{1}\right)\ge 0$.

eyiliweyouc

Expert

Step 1
$f\left(x\right)=\left\{\begin{array}{ll}-4& x\in \left[0,2\right]\\ -2& x\in \left[2,4\right]\\ 0& x\in \left[4,6\right]\end{array}$
$g\left(x\right)=\left\{\begin{array}{ll}5& x\in \left[0,1\right]\\ 3& x\in \left[1,3\right]\\ 1& x\in \left[3,5\right]\\ 0& x\in \left[5,6\right]\end{array}$
Step 2
$q\left(x\right)=\left\{\begin{array}{ll}1& x\in \left[0,1\right]\\ -1& x\in \left[1,2\right]\\ 1& x\in \left[2,3\right]\\ -1& x\in \left[3,4\right]\\ 1& x\in \left[4,5\right]\\ 0& x\in \left[5,6\right]\end{array}$
This example could be made continuous and strictly monotone with some tweaking.

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