Consider the function $f(x)=x^3+3x^2-45x+4.$

$($a$)$ Find the intervals on which $f$ is increasing or decreasing.

$($b$)$ Find the local maxima and local minima of $f.$

$($c$)$ Find the intervals on which $f$ concave upwards or concave downwards.

$($d$)$ Find the inflection points of $f.$