Final answer:
The student needs to apply the Closed Interval Method to find the absolute maximum and minimum of f(x) = x/(x² - x + 9) on the interval [0, 9] by first finding the derivative, then checking critical points and endpoints.
Step-by-step explanation:
The student seeks the absolute maximum and absolute minimum values of the function f(x) = x/(x² - x + 9) on the interval [0, 9]. To find these values, we apply the Closed Interval Method which involves several steps:
- Calculate the derivative of the function f'(x) to find critical points.
- Check the value of the function at the critical points and the endpoints of the interval.
First, differentiate f(x) to get f'(x). We find that f'(x) = -x² +9/(x² - x + 9)². Setting the derivative equal to zero, we solve for x to find any critical points within the interval.
Next, evaluate the function at the critical points and the endpoints of the interval, which are x = 0 and x = 9 in this case. The function evaluated at these points gives us the potential maximum or minimum values.
Comparing these values will give us the absolute maximum and absolute minimum of the function on the given interval.