Final answer:
The absolute maximum value of the function f(x) on the interval [-3, 5] is -49, and the absolute minimum value is -367, calculated by finding critical points and evaluating the function at these points and the interval endpoints.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = 6x³ - 18x² - 144x + 1 on the interval [-3, 5], we first need to find the critical points of the function within this interval and evaluate the function at these points and at the endpoints of the interval.
First, let's find the derivative of f(x) to locate critical points:
f'(x) = 18x² - 36x - 144.
Set the derivative equal to zero to find critical points:
18x² - 36x - 144 = 0.
We can factor out a 6 to make the equation simpler:
3x² - 6x - 24 = 0.
Now solve for x:
(3x + 6)(x - 4) = 0,
So the critical points are x = -2 and x = 4. Since both of these points are within our interval, we will evaluate the function at x = -3, x = -2, x = 4, and x = 5.
f(-3) = 6(-3)³ - 18(-3)² - 144(-3) + 1 = -162,
f(-2) = 6(-2)³ - 18(-2)² - 144(-2) + 1 = -49,
f(4) = 6(4)³ - 18(4)² - 144(4) + 1 = -367,
f(5) = 6(5)³ - 18(5)² - 144(5) + 1 = -241.
Comparing these values, we find that the absolute maximum value of f(x) on the interval is -49 and the absolute minimum value is -367.