Final answer:
To find the absolute maximum and minimum values, we first calculate the function's critical points where the derivative is zero and compare the function's values at these points with those at the interval's endpoints.
Step-by-step explanation:
Finding the Absolute Maximum and Minimum
To find the absolute maximum and absolute minimum values of the function f(x) = x³ - 9x² + 5 on the closed interval [-4, 7], we first need to look for critical points where the derivative of the function equals zero or is undefined and then evaluate the function at the endpoints of the interval.
First, we find the derivative of the function: f'(x) = 3x² - 18x.
Second, we set the derivative equal to zero to find critical points: 3x² - 18x = 0. Factoring out a 3x gives 3x(x - 6) = 0, so the critical points are at x = 0 and x = 6.
Third, we evaluate the function at these critical points and at the endpoints of the interval: f(-4), f(0), f(6), and f(7).
Next, by comparing these values, we can determine which are the largest (absolute maximum) and smallest (absolute minimum).
Once these steps are completed, the highest and lowest values obtained are the absolute maximum and minimum of f(x) on the interval [-4, 7].
Graphing Considerations
To graph the function, label the graph with f(x) and x. Scale the axes to accommodate the maximum x (which is 7) and the maximum value f(x) computed from the critical points and endpoints. Since we are given the function f(x) and not f(x) = 10, we ignore the provided function and scale the graph based on our computed values instead.