Let $f: [a, b] \rightarrow \mathbb{R}$ be a function, continuous on$[a, b]$ and twice differentiable on $(a, b)$. If $f(a) = f(b)$ and $f'(a) = f'(b)$. Then find the least number of roots of the equation $f''(x) - \lambda (f'(x))^2 = 0$, for any real $\lambda?$
I know that I need to imagine some function, then use Rolle's Theorem, but can't think of the function. I need some insight, how should I start?