Size of a population is given by $N(t)=10+2e^{-0.3t}sin t$
The size is measured in millions and time is in weeks.
Plotting the graph in Maple gives me this image:
How do I get the upper and bounds on the size of population using minima and maxima concept?
First you need to find derivative $N'(t)=-0.6e^{-0.3t}\sin t+2e^{-0.3t}\cos t$ then you need to find all solutions for $N'(t)=0$ which is equivalent to solving $-0.6\sin t+2\cos t=0$ or $\tan x=\frac{10}{3}$. The solutions of this equation are $t=\arctan{\frac{10}{3}}+n\pi, n= 0, 1 ,2, ...$. These are the points that you need to check for minimum and maximum. The most interesting for you are probably the first few solutions as $N(t)$ tends to $10$ as $t$ tends to infinity.