A postdoc tries to fix the grad student$$s mistake by proposing the following fixed point form:

x $=$x

$3+$

$2$

$3$

x

$2+$

$2$

$3$

$,$

which inspires the iteration scheme

xk$+1=$x

$3$

k $+$

$2$

$3$

x

$2$

k $+$

$2$

$3$

$.$

Does this fixed point iteration converge for all initial guesses sufficiently close to the root of

f$?$