Task$4.$

Do the codes.

Let $f(x)$ be a function of $x.$

$f(x)=x^5+2\mathrm{.}5x^4-2x^3-6x^2+\frac{x}{2}+2$

a$.$ Find the actual roots of $f(x)$ and print them.

b$.$ Plot the function for $-2\mathrm{.}5x1\mathrm{.}5,$ also point out the the found roots in the plot

c$.$ The following $g_1(x)$ is given which is derived from Eq$(4\mathrm{.}1),$

Use Contraction Mapping Theorem and calculate the value of $$ for the given $g(x)$

$g_1(x)=\frac{1}{2}(-x^5-2\mathrm{.}5x^4+2x^3+6x^2-2)$

d$.$ Compute the convergence$/$divergence table using all the calculated roots for the given $g_1(x)$ and prove the whole $g_1(x)$ is divergent

Given,

$g_2(x)=\sqrt[\phantom{2}]{\frac{1}{6}(x^5+2\mathrm{.}5x^4-2x^3+\frac{1}{2}x+2)}$

$g_3(x)=\sqrt[4]{\frac{1}{2\mathrm{.}5}(-x^5+2x^3+6x^2-\frac{1}{2}x-2)}$

e$.$ Derive $2$ more separate $g_4(x)$ and $g_5(x)$ from the given $f(x).$ Implement $g_2(x),g_3(x),g_4(x)$ and $g_5(x).$

f$.$ Apply Fixed Point Method on the $g_2(x),g_3(x),g_4(x)$ and $g_5(x).$ and find the approprate roots, show $20$ iterations for each $g(x)$ for $x_0=0\mathrm{.}8$

and show the convergence table using data from each iteration

g$.$ Plot the $g(x)s$ where actual roots were found along with $f(x).$

$[]$ #$4$a this cell should print

$[]$ #$4$b This cell should print plot a graph

$[]$ #$4$c This cell should print

$[]$ #$4$d this cell should print

$[]$ #$4$e This cell have no outputs

$[]$ #$4$f This cell should print

$[]$ #$4$g$\}\backslash$mathrm$\{$ this cell should plot a graph. Do not plot those g$($x$)$ which will not converge.