Bessel Function and roots

Question

How would you prove that if $p_k$ and $p_{k+1}$ are two consecutive zeros of the Bessel function $J^{(0)}(t)$ then there exist a $t_1$ such that $p_k\leq t_1 \leq p_{k+1}$ and $J^{(1)}(t_1) = 0$?. Is there any generalization of this?

Answer 1

Rolle's theorem says that if $f$ is differentiable on $(a,b)$ (and continuous on $[a,b]$) and $f(a) = 0 = f(b)$, then $f'(c) = 0$ for some $c\in(a,b)$. Letting $f(x) = J_0(x)$, we see that $f(p_k) = 0 = f(p_{k+1})$. But then this says that $f'(c) = 0$ for some $c\in (p_k,p_{k+1})$. The standard identity $J_0'(x) = -J_1(x)$ then yields the result.