Final answer:
The function f(x) = x³ - 12x has an absolute maximum value of 65 at x=5 and an absolute minimum value of -16 at x=2 on the interval [-3,5]. The options provided in the question do not match these values, suggesting an error.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = x³ - 12x on the closed interval [-3,5], we can follow these steps:
- Find the critical points of the function by solving f'(x) = 0.
- Evaluate the function at the critical points and at the endpoints of the interval.
- Determine which of these values are the highest (absolute max) and lowest (absolute min).
The first derivative of the function is f'(x) = 3x² - 12. Setting this to zero and solving for x, we find that the critical points are at x = -2 and x = 2. Evaluating the function at these points and at the endpoints -3 and 5 gives us:
- f(-3) = -27 + 36 = 9
- f(-2) = -8 + 24 = 16
- f(2) = 8 - 24 = -16
- f(5) = 125 - 60 = 65
The absolute maximum value is f(5) = 65, and the absolute minimum value is f(2) = -16. Thus, the correct answer is Absolute Max: (5, 65), Absolute Min: (2, -16), which was not listed in the provided options, indicating a possible error in the question itself.