Consider the function

$f(x)=\sqrt[3]{x}(x-4).$

$($a$)$ Find the domain.

$($b$)$ Find the $x-$ and $y-$intercepts.

$($c$)$ Is the function even? Is the function odd?

$($d$)$ Find the vertical and horizontal asymptotes $($if any$).$

$($e$)$ Find the intervals of increase and decrease.

$($f$)$ Find the local maximum and local minimum points $($if any$).$

$($g$)$ Where is the graph of $f$ concave up$?$ Where is it concave down?

$($h$)$ Find the inflection points $($if any$).$

$($i$)$ Using this information, sketch a graph of this function.