A demographic study of a town indicates that P(r) hundred people live r miles from the centre of the town, where P(r)=5(3r+1).
r^2+r+2

a)What is the population at the centre of the town?
b)for what values of r is P(r) increasing? For what values is P(r) decreasing?
c)At what distance from the town centre is the population the largest? And what is the largest population?

a) At the centre of the town, r = 0, so, find P(0)

b) (I'm not sure if that's what's expected for you to do, but this is how I would work it out and this gives the answer)

Find the derivative of P(r), set to zero and solve for r.

You'll get two values of r, but you have to deduce which one is the correct one. At this value of r, the function is neither increasing nor decreasing.

You can now either:
- Check the gradient on the right and the left of the curve to know where it's increasing and where it's decreasing, or
- Find the second derivative and find whether the point you found is a maximum or a minimum. A maximum will indicate the function is initially increasing and after the value of r you got, it becomes a decreasing function. The opposite is also true.

c) If the value of r you got above is at maximum (you can check by finding the second derivative of P)

Use that r in P(r)

I cannot see how it can be a minimum if it asks for the maximum population.