Final answer:
The function f(x) = x³ - 9x has a positive average rate of change over the interval (√3, 20), where the derivative f'(x) = 3x² - 9 is greater than zero.
Step-by-step explanation:
To determine over which interval the function f(x) = x³ - 9x has a positive average rate of change, we need to look at the intervals where the function is increasing. The average rate of change of a function over an interval [a, b] is calculated by the formula ∆f/∆x = (f(b) - f(a))/(b - a). A positive value indicates that the function is increasing on average over that interval.
To find such intervals, we can analyze the function's derivative, f'(x) = 3x² - 9. This derivative will be positive where the function is increasing. Setting f'(x) > 0, we get 3x² - 9 > 0, which simplifies to x² > 3, so x > √3 or x < -√3. Therefore, the intervals over which the function has a positive average rate of change are (-∞, -√3) and (√3, ∞). However, to determine a specific interval that falls completely within the given domain of 0 ≤ x ≤ 20, we should focus on the interval (√3, 20).