Consider the equation x$32$x $5=0$

$($a$)$ Use the Intermediate Value Theorem to prove that this equation has a root in

the interval $[2,3].$

$($b$)$ Starting with x$0=2\mathrm{.}5,$ use Newton$-$Raphson method to find this root.

$($c$)$ Continue iterations until two successive approximations differ by less than

$0\mathrm{.}0005$