maths for social science
posted by Jade .
. A project management consultant estimated that if a particular project was completed, that t years after its date of completion, P thousand persons would be direct beneficiaries of the project, where
P (t) = t3/3  5t2 +20t, 0<t<12
For what value of t will the maximum number of people receive direct benefits?

to find where P(t) is a maximum, find where its derivative is zero:
P'(t) = t^2  10t + 20
t = 5 ± √5
That is where P(t) has a min or a max. To determine which, check
P''(t) = 2t  10
At t = 5  √5, P'' < 0, so P(t) has a max.
So, at t = 5  √5 = 2.764, P will be at a maximum.