Consider the equation below. $($If an answer does not exist, enter DNE.$)$

$f(x)=x^3-9x^2-21x+4$

$($a$)$ Find the interval$($s$)$ on which $f$ is increasing. $($Enter your answer using interval notation.$)$

Find the interval$($s$)$ on which $f$ is decreasing. $($Enter your answer using interval notation.$)$

$($b$)$ Find the local minimum and maximum values of $f.$

local minimum value

local maximum value

$($c$)$ Find the inflection point.

$(x,y)=()$

Find the interval$($s$)$ on which $f$ is concave up$.($Enter your answer using interval notation.$)$

Find the interval$($s$)$ on which $f$ is concave down. $($Enter your answer using interval notation.$)$