What is the average speed of an object that is still at #t=0# and accelerates at a rate of #a(t) =3t-1# from #t in [2, 3]#?
It's speed is 0
First Newton's law says:
Any object in which no force is aplyed ( no acceleration) will remain still, or moving at a constant speed.
If at t=0 your object was still and till the clock is at t=2 there is no acceleration ( cause that acceleration only aplyes for time between 2 and 3 seconds) there is no force aplyed to the object at t=0 so it's speed won't change till an acceleration is aplyed, by your equations this acceleration starts being aplyed at t=2
So BEFORE t=2 the object will remain still, once the clock is at t=2 it will start accelerating following your a(t) equation