I need to figure out what the graph of f(x)= 1 / (2-3x)^3 near x=2/3 would look like.

f'=-3(-3)/(2-3x)^4

f"=-36/(2-3x)^5

now look at f', and f" at each side of 2/3, say at .5, and at 1.0

I will do the 1.0 side...
f=1/(2-3)^3=1
f'=9/(2-3)^4= positive 9, high slope
f"=-36/(2-3)^5=36 positive, curving up

now the .5 side.
f=1/(2-1.5)^3=8
f'=9/(2-1.5)^4=9*64 positive, going upwards
f"=-36/(.5)^5=-1152 negative curving down,slope is reducing.

Now graph it on your calculator and take a look