http://www.math.rutgers.edu/~greenfi...fstuff/w3N.pdf
To that problem

I have

a) increasing for [0,C) and (D,E] since f'(x) is positive there, and so, decreasing for (C,D). Local extrema would just be C as a local max? (Not sure though since it isn't continuous)

Yes, local max at C. But what happens at D?
b) concave up=f''(x)>0 for [0,A)U[B,C)U(C,E] and concave down only for (A,B) since that is the only place where slope is negative. So point of inflection would be both A and B?

Yes.
c) Not sure how I could draw the graph....Do I just draw anything that fits those specifications

Yes!
or how can I know where things start and stop?

The only information you have cincerns the derivative. So f can only be determined up to a constant (like a constant of integration).