this is probably a better way of setting the answer out:
population increases most rapidly at point of inflextion, ie, when d^2P/dt^2 = 0
d^2P/dt^2, ie the second derivative = 12 - 6t
12 - 6t = 0
6t = 12
therefore, t = 2
check for concavity change
when t = 3, 2nd derivative = -6 . . ...