The roots of quintic polynomial

Question

I have some region $[a,b],$ and the polynomial $P(x)$ in it $(\deg P(x)=5)$: Given these conditions

1. $P(a)\cdot P(b)\lt0$
2. $P'(x)\gt 0$ or $P'(x)\lt 0$ when $x\in (a,b)$
3. $P'(a) = P'(b) = 0$

I know that in these conditions there is only one root in the region [a,b], and I can find the root with the Bisection method. But I know that these method is very slow and I want to use Newton's method. I want to find some x0 initial point for the Newton's method, for which the method will always converges to the root.