$x_1=(x_0)=x_0-\frac{4f_0}{f_0^{\text{'}}+-[9f_0^{\text{'}2}-12f_0{f}_0^{\text{'}\text{'}}{]}^{\frac{1}{2}}}.$

$($i$)$ Write a Maple procedure which takes as its arguments a function $f,$ an initial estimate

for a root $x_0,$ a number of steps $n$ and a boolean called display. The procedure

should use the function $$ from part $($d$)$ to produce a sequence of iterates $x_1,x_2,$dots,$x_n,$

such that $x_{j+1}=(x_j).$ In each step, the sign that gives the denominator the larger

magnitude should be chosen. The procedure should return $x_n$ as its result. If display

is true, the procedure should display $|f(x)|$ at each step using scientific notation $($use

$\%10\mathrm{.}3$ e with printf$).$ If display is false, no information should be displayed while

the procedure is running.

$($ii$)$ Test your procedure for the case

$f(x)=10x^4-61x^3+13x^2+122**x-66,$

with the initial estimate $x_0=5\mathrm{.}0.$ Perform a further test using an arbitrarily chosen

function $($but not a polynomial$).$ Estimate the rate of convergence in both cases.