The function $\phi(x) \in C^p([a,b])$ and $x_0 \in [a,b]$ meet the conditions for convergence of the fixed-point iteration method, that is: $$\phi([a,b]) \subseteq [a,b] \\ \phi'(x) < 1 \mspace{5mu} ,\mspace{15mu} \forall x \in [a,b]$$
Let $\alpha \in ]a,b[$ be the solution of $\phi(\alpha) = \alpha$. Prove that if: $$\phi'(\alpha)=\phi''(\alpha)=...=\phi^{(p-1)}(\alpha) = \beta$$
Then: $$\beta=0$$
I thought about using: $$\lim_{k \rightarrow +\infty}\left(\frac{x_{k+1}-\alpha}{x_k-\alpha}\right) = \phi'(\alpha)$$ But it got me nowhere. I also thought about using the Taylor polynomial of $\phi$ around the point $\alpha$ but it didn't seem to take me anywhere either.