Q:  Given ATC=3Q²-2Q+4/Q, find FC and VC.

A: There are several different ways to solve this problem, since we know several different relations among the cost curves:
TC=VC+FC, FC=TC(when Q=0),  ATC=TC/Q, AVC=VC/Q, AFC=FC/Q, ATC=AFC+AVC, and MC=dTC/dQ=dVC/dQ.
The easiest way to proceed is to first calculate TC and then use the the fact that FC=TC(when Q=0) and finally solve for VC=TC-FC

To find TC:  ATC=TC/Q so TC=ATC*Q.  TC = (3Q²-2Q+4/Q)*Q = 3Q³-2Q²+4

To find FC:  FC=TC(when Q=0).  FC = 3(0)³-2(0)²+4 = 0-0+4 = 4

To find VC:  VC=TC-FC.  So VC = 3Q³-2Q²+4-4 = 3Q³-2Q²