Show that $\exists m \in(k,\ell):f''(m)+f(m)=0$ Let $f$ be twice differentiable on $\Bbb R$ and $f'(x)\not=0$, $f'(k)=f(\ell)$ and $f'(\ell)=f(k)$. Show that $\exists m \in(k,\ell):f''(m)+f(m)=0$.
The only thing that I was able to do is to show that $\exists n \in (k,\ell):f'(n)-f(n)=0$. I'd like a hint.

Hint: consider
$$
h(x)=(f(x))^2+(f'(x))^2
$$
and see what the mean value theorem implies.
