$($a$)$ Use the Intermediate Value Theorem to show that x$4$ x $1=0$ has a root

between x $=1$ and x $=2.$

$($b$)$ Find this root using Newton$-$Raphson method with x$0=1\mathrm{.}5$

$($c$)$ Stop when $|$xi $$ xi$1|<0\mathrm{.}0005$

$($d$)$ For each iteration, show:

$$ The value of xi

$$ The value of f $($xi$)$

$$ The absolute difference $|$xi $$ xi$1|$