Shutting off production in short term
I think I understand the problem now, your AVC(Q) is wrong. Your fixed costs are 4000, because it's what the firm will pay after it stops production. VC=TC-4=Q^3+2. Thus AVC=Q^2+2/Q. Now you should use your calculus in order to find where AVC has local positive minimum (because it doesn't have global minimum due to its hyperbolic nature).