Final answer:
The absolute maximum and minimum of a function on a given interval are found by evaluating the function's critical points and endpoints, and then comparing the values. In this case, it involves taking the derivative of f(x) = x³ - 12x² - 27x + 4 and finding where it equals zero within the interval (-2, 0), in addition to testing the endpoints.
Step-by-step explanation:
To find the absolute maximum and absolute minimum of the function f(x) = x³ - 12x² - 27x + 4 on the interval (-2, 0), we need to follow several steps. First, we calculate the critical points of the function by finding the derivative of f(x) and setting it equal to zero. Secondly, we evaluate the function at the critical points and at the endpoints of the interval. Lastly, we compare these values to determine which is the absolute maximum and minimum within the given interval.
The derivative of the function is f'(x) = 3x² - 24x - 27. Setting this derivative equal to zero, we solve for x to find the critical points. However, since the interval is (-2, 0), we will only consider critical points that lie within this interval. After the critical points are determined, we then evaluate f(x) at these points as well as at the endpoints x = -2 and x = 0. The highest value obtained will be the absolute maximum, and the lowest value will be the absolute minimum on the interval (-2, 0).