Final answer:
The function f(x) = x² - 7x is increasing on the interval (7/2, 20], and it is decreasing on the interval [0, 7/2).
Step-by-step explanation:
The function f(x) = x² - 7x is a quadratic function. To determine the interval(s) where the function is increasing or decreasing, we need to find the critical points of the function. The critical points occur where the derivative of the function is equal to zero or does not exist.
To find the derivative, we can use the power rule. The derivative of x² is 2x, and the derivative of -7x is -7. So the derivative of f(x) = x² - 7x is f'(x) = 2x - 7.
To find the critical points, we set f'(x) equal to zero and solve for x:
2x - 7 = 0
2x = 7
x = 7/2
Since the domain of the function is 0 ≤ x ≤ 20, we need to check the behavior of the derivative at the endpoints as well. Evaluating f'(0) and f'(20), we find that f'(0) = -7 and f'(20) = 33.
Therefore, the function f(x) = x² - 7x is increasing on the interval (7/2, 20], and it is decreasing on the interval [0, 7/2).