Final answer:
To find the absolute maximum value of the function f(x) = x^4 - 8x^3 + 16x^2 - 8 on [-5, -1], calculate f(x) at the critical points and endpoints of the interval, then select the highest value.
Step-by-step explanation:
To determine the absolute maximum value of the function f(x) = x⁴ - 8x³ + 16x² - 8 on the closed interval [-5, -1], we need to evaluate the function at the critical points within the interval as well as at the endpoints.
The critical points are found by taking the derivative of f(x) and setting it equal to zero, however, since the domain is restricted to the interval [-5, -1], we only need to consider critical points within this range. Additionally, we'll evaluate the function at x = -5 and x = -1 to find the absolute maximum value on the given interval.
To find the critical points:
- Calculate the first derivative: f'(x) = 4x³ - 24x² + 32x.
- Solve f'(x) = 0 for x within the interval [-5, -1].
After finding the critical points, calculate f(x) for each critical point and at the endpoints of the interval. The highest value obtained from these calculations will be the absolute maximum value of f(x) on the interval [-5, -1].