For $f$ defined on $[0,1]$ twice differentiable, and two continuous function defined on $[0,1]$ named $p(x)$, $q(x)$, satisfying
$f''+pf'+qf=0$, where $q\leq0$ and $f(0)=f(1)=0$,
I haven't got an idea about the question. Any hint will be appreciated.
Edited: for $q(x_0)<0$ where $f'(x_0)=0$ the problem can easily be solved. So now I'm mainly concerned about $q(x_0)=0$. It appears on my textbook, so i guess it's probably true. Counter examples are also welcomed.