Consider the sequence \(\{x_n\}_{n\geq 0}\) defined by the recurrence relation…

Consider the sequence $\{x_n\}_{n\geq 0}$ defined by the recurrence relation $x_{n+1}=cx_n^2-2$, where $c>0$.

For which of the following values of $c$ does there exist a non-empty open interval $(a,b)$ such that the sequence $x_n$ converges for all $x_0$ satisfying $a<x_0<b$?

(i) 0.25 (ii) 0.35 (iii) 0.45 (iv) 0.5