Fill in the gaps in the function newton$\_$root$\_$finder below so that it returns the approximation xn$+1$ for a root of the given function f $=$ f with derivative f$=$ f$\_$prime found using Newton's method with initial approximation x$0,$ using the given $=$ tolerance value applied to both the function error and the approximation change, ie$,$ iteration stops when both of the following are true:

$|$f$($xn$+1)|<$and$|$xn$+1$xn$|<$

provided that this happens without exceeding the specified maximum number of iterations max$\_$iterations. If the specified maximum number of iterations is exceeded the function should raise a RuntimeError with the message "Max iterations reached without convergence in newton$\_$root$\_$finder".

For example, if the function is tested with the code below:

f $=$ lambda x: x$**3+$ x $+1$

f$\_$prime $=$ lambda x: $3*$ x$**2+1$

x$0=-1$

tol $=0\mathrm{.}5$e$-3$

max$\_$iterations $=20$

try:

root $=$ newton$\_$root$\_$finder$($f$,$ f$\_$prime, x$0,$ tol, max$\_$iterations$)$

print$($f$"$Root found $\{$root:$\mathrm{.}6$f$\}(6$ dps$)")$

except RuntimeError as err:

print$($err$)$

the test code should print:

Root found $-0\mathrm{.}682328(6$ dps$)$