Final answer:
The absolute maximum occurs at x = -2 and the absolute minimum occurs at x = 3 when evaluating the function f(x) = x^3 - 5x^2 + 3x + 12 on the interval [-2, 4].
Step-by-step explanation:
To find the absolute maximum and absolute minimum values of the function f(x) = x^3 - 5x^2 + 3x + 12 on the interval [-2, 4], we can start by finding the critical points. These are the points where the derivative of the function is zero or undefined. In this case, the derivative of f(x) is f'(x) = 3x^2 - 10x + 3.
Setting f'(x) equal to zero and solving for x gives us x = -1 and x = 3 as the critical points. We also need to check the endpoints of the interval. Evaluating f(-2), f(4), and f(x) at the critical points, we find that the absolute maximum occurs at x = -2 and the absolute minimum occurs at x = 3.