Question

The amount A(t) of qa certain item produced in a factory is given by A9t) = -4000 + 48(t-3) - 4(t-3)^3
where t is the number of hours of production since the beginning of the workday at 8:00 am. At what time is the rate of production increasing most rapidly?

Answer 1

Maximize the first derivative, which means look for boundaries and critical points in the second derivative.

Answer 2

ok, so I find the inflection points... and to find boundaries?

Answer 3

Don't worry about the boundaries in this case. Yes, in a way you are finding inflection points, but that is not the best way to think of it. What you really want to do is maximize the derivative of the function. You do that by finding critical points of the second derivative, which are inflection points of the function, but you should still use the second derivative test, to find inflection points of the first derivative, by finding critical points of the third derivative.
I'm sorry, that wasn't exactly my most eloquent prose...

Answer 4

ok, so I got as far as finding the CNs for the second deriviative, in which I got t = 3, and that's it. Now I have to find them for the 3rd derivative?

Answer 5

If there's only one, you probably have your answer. Using the third derivative would be for using the second derivative test on the first derivative.

Answer 6

I got 11 am.

Answer 7

I haven't solved the problem. If you did the steps properly, you will have your solution.