Answer
(a) The population is increasing when $0 \lt P \lt 4200$
(b) The population is decreasing when $P \gt 4200$
(c) The population is in equilibrium when $~P = 0~$ or $~P= 4200$
Work Step by Step
(a) The population is increasing when $\frac{dP}{dt} \gt 0$
If $P \gt 0$, then $2P\gt 0$
If $P \lt 4200$, then $(1-\frac{P}{4200})\gt 0$
Therefore, the population is increasing when $0 \lt P \lt 4200$
(b) The population is decreasing when $\frac{dP}{dt} \lt 0$
If $P \gt 0$, then $2P\gt 0$
If $P \gt 4200$, then $(1-\frac{P}{4200})\lt 0$
Therefore, the population is decreasing when $P \gt 4200$
(c) The population is in equilibrium when $\frac{dP}{dt} = 0$
If $P = 0$, then $2P = 0$, and $\frac{dP}{dt} = 0$
If $P = 4200$, then $(1-\frac{P}{4200}) = 0$, and $\frac{dP}{dt} = 0$
Therefore, the population is in equilibrium when $~P = 0~$ or $~P= 4200$
