For what range of t is R(t) increasing?

Question

R(t) = 15t^2 - 1/3t^3

Answer 1

The first derivative shows you where the max and min values are and shows you what the slope of your function is at any time t. To start, we need to find the first derivative and set it to 0. So:

R'(t) = 30t - t^2
0 = 30t - t^2
0 = t(30 - t)
t = 0, 30

The first derivative gives you the slope of a function. So if we analyze this, we can see what's going on (1) to the left of 0, (2) between 0 and 30, and (3) to the right of 30.

Pick something to the left of 0 - say, -100000000. Plugging this in to the first derivative gives you something positive (it doesn't really matter what the value is), which means that the slope is positive and the function is thus increasing on (-inf, 0).

Pick something between 0 and 30 - say, 1 (it's easy to work with). Plug this in and we see that you also get something positive, which means that our function is increasing here as well. So, we're increasing now from (-inf, 30).

Plug in something to the right of 30 - say, 100000000. Here, you can see that the derivative becomes negative, which means the function has a negative slope or is decreasing after 30.

So, your answer is (-inf, 30) - you can't include 30 in this because that's a relative maximum for this function.