Final answer:
To find the absolute maximum and minimum values of f(x) = 2x³ - 3x² - 72x on the interval [-4, 5], we find the critical points of the function and evaluate it at those points and the endpoints of the interval.
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = 2x³ - 3x² - 72x on the interval [-4, 5], we first need to find the critical points of the function. We take the derivative of f(x) with respect to x, which gives us f'(x) = 6x² - 6x - 72. Setting f'(x) equal to zero and solving for x, we get x = -3 and x = 4.
Next, we evaluate the function at these critical points and at the endpoints of the interval [-4, 5]. We have f(-4) = -304, f(-3) = -315, f(4) = -304, and f(5) = -205.
Therefore, the absolute maximum value of f(x) on the interval [-4, 5] is -205 and it occurs at x = 5, while the absolute minimum value of f(x) on the same interval is -315 and it occurs at x = -3.