How to find the maximum peak of $\phi$ on $[0,l]$.

Question

You are given, $\phi(r)=a_1J_0(\alpha r)+a_2J_0(\beta r)$, where $a_1$ and $a_2$ are constants so that $a_1+a_2=1$, and $r\in[0,\infty)$. $\alpha$ and $\beta$ are parameters, which are positive. If there is some $l>0$ such that $\phi(l)=0$ and $\phi$ is non negative on $[0,l]$. How to find the maximum peak of $\phi$ on $[0,l]$. Here $J_0$ denotes the Bessel function of the first kind.

Answer 1

As usual, the maximum occurs at a critical point. We want to solve $$ \phi'(r) = -\alpha a_1 J_1(\alpha r) - \beta a_2 J_1(\beta r) = 0$$ for $r$, which will generally have to be done numerically.