Consider the following four methods for calculating $21$

$4,$ the fourth root of $2.$

$($i$)$ Bisection Method applied to f$($x$)=$ x

$42$

$($ii$)$ Secant Method applied to f$($x$)=$ x

$42$

$($iii$)$ Fixed Point Iteration applied to g$($x$)=$ x

$2+$

$1$

x$3$

$($iv$)$ Fixed Point Iteration applied to g$($x$)=$ x

$3+$

$1$

$3$x$3$

$($a$)$ Rank them for speed of convergence, from fastest to slowest. Be sure to give

reasons for your ranking.

$($b$)$ Are there any methods that will converge faster than all above suggestions?