Question

Over which interval does the function h(t) = (t³)² - 5 have a negative average rate of change?

1) (-[infinity], 0)
2) (0, [infinity])
3) (-[infinity], -1) ∪ (1, [infinity])
4) (-1, 1)

Answer 1

The interval over which the function has a negative average rate of change is (-∞, 0)

Step-by-step explanation:

To determine the interval over which the function has a negative average rate of change, we need to find where the function is decreasing. The average rate of change is negative when the function is decreasing.

We can find the points where the function is decreasing by finding where the derivative of the function is negative. Thus, we need to find where the derivative of the function, h'(t), is negative.

First, let's find the derivative of the function: h'(t) = 2(t³)(3t²) = 6t⁵.

Now, we need to find where 6t⁵ is negative. Since t³ is always positive, 6t⁵ will be negative only if t is negative. Therefore, the interval over which the function has a negative average rate of change is (-∞, 0) or option 1.