A double-unimodal function is an analytic function that is positive and unimodal on $(0,r)$ and negative and unimodal on $(r,\infty)$, $f(0)=f(r)=f(\infty)=0$.
Suppose $f$ and $g$ are double-unimodal functions with simple roots $r_f$ and $r_g$. Suppose also that $f(a)=g(a)$ for only one value $a$. Suppose $0<a<r_f<r_g<\infty$. I would like to know where the largest root of their sum is. For example,
Suppose that $(f+g)(b)=0$ and $f+g$ is negative from $(b,\infty)$. Under what conditions is $b>(r_f+r_g)/2$?