A flu epidemic breaks out in a city. The fraction of the population that is infected at time t, in weeks, is given by the function f(t)=64t/(8+t)^3. What is the largest fraction of the population that is infected during the first 10 weeks? Assume that t=0 when the epidemic starts.

Ok.. I think I have my method correct for this, but I am not exactly sure, so can someone verify my method and calculations to see if I have the correct answer? Thank you very much in advance

M_t=f'(t)=[64(8+t)^3-(64t)(3(8+t)^2)(1)]/((8+t)^3)^2

=[64(t^3+21t^2+144t+320)]/(8+t)^6

Critical Numbers

f'(t)=0 when t = -5 or t = -8

f'(t)=undefined when t = -8

Reject answers above because all negative.. time cannot be negative

In this case, domain is 0<=t<=10.. therefore

Test end points

f(0)=0

f(10)=640/5832=0.1097

Therefore the largest fraction of the population that is infected during the first ten weeks is 0.1097 when it is the tenth week.