Find all critical numbers for the following function. Then use the second$-$derivative test on each critical number to determine whether it leads to a local maximum or minimum.

$f(x)=-2x^3-3x^2+12x+2$

Select the correct choice below and, if necessary, fill in the answer box$($es$)$ to complete your choice.

A$.$ The critical number$($s$)$ is$/$are $x=q,$ There is a local maximum at $x=$ There is a local minimum at $x=($Use a comma to separate answers as needed.$)$

B$.$ The critical number$($s$)$ is$/$are $x=q,$ There is a local maximum at $x=q,$ There is no local minimum.

$($Use a comma to separate answers as needed.$)$

C$.$ The critical number$($s$)$ is$/$are $x=$ There is no local maximum. There is a local minimum at $x=$

$($Use a comma to separate answers as needed.$)$

D$.$ The critical number$($s$)$ is$/$are $x=$ There is no local maximum and no local minimum.

$($Use a comma to separate answers as needed.$)$

E$.$ There are no critical numbers.