Let $\phi(x)$ is a polynomial function, with real coefficients. Let $\alpha$ and $\beta$ be any two consecutive roots of $\phi(x)=0$, then prove that there lies a root of $\phi'(x)+\lambda \phi(x)=0$ in the interval $(\alpha, \beta)$. (here $\lambda$ is a fixed constant).
This is a solution-verification type post. This question has already been answered here. If you have another answer to this question, kindly post it here
My approach:
We need to prove that there exists $x\in(\alpha,\beta), $ such that $\phi'(x)=-\lambda \phi(x)$.
Since $\phi(\alpha)=0$ and $\phi(\beta)=0$ and $\phi(x)$ is a polynomial, hence $\phi(x)$ attains a maxima or a minima in $(\alpha, \beta)$; hence, $\phi'(c)=0$ for some $c\in(\alpha, \beta)$.
Now, I don't think the above statement should be true, because for example, what if $\beta$ was a repeated root of $\phi$(even no. of times repeated) and $\phi'$(odd no. of times repeated, obviously). something like this
unfortunately, I haven't been able to generate a specific polynomial that behaves in this way. so I'm stuck.
My question-
could someone comment on why my approach was wrong? more specifically, I laid out a "blueprint" for a counter-example(the blue image above) but wasn't able to find an actual counter-example(because such a counter-example doesn't exist). even after solving this problem, I have not been able to entirely convince myself, why such a counterexample shouldn't exist. so that's my question, why can such a counterexample exist not exist?