Prove that the iterative scheme
$$X_{r+1} = g(X_r) = e^{X_r^{2}-2X_r}$$
with a suitable starting point, converges to the root in $[0.4,0.6]$, by showing that $g$ is a contraction mapping on this interval.
Compute the root in $[0.4,0.6]$ to two decimal places
Why the iterative scheme would not be appropriate to use to try to find the larger root?
$g(x) = e^{x^2 - 2x}$
$|g'(x)| = |(2x - 1)e^{x^2 - 2x}| \leq 0.2 \times e^{-0.64} < 1$
By Banach fixed point theorem, the fixed-point iteration converges with any initial guess in $[0.4, 0.6]$.
If the root is larger, then $|g'(x)|$ can be very large in a neighborhood of the root. The fixed-point iteration may fail.