Final Answer:
To find the absolute maximum and minimum of a function, identify critical points by setting the derivative equal to zero and solve for x. Additionally, examine the endpoints of the function's domain. The highest and lowest values among these critical points and endpoints represent the absolute maximum and minimum, respectively.
Step-by-step explanation:
To determine the absolute maximum and minimum of a function, start by finding the critical points. These are the values of x where the derivative of the function is equal to zero or undefined. Set the derivative equal to zero and solve for x. Once you have the critical points, evaluate the function at each of these points and also at the endpoints of the function's domain. The highest value obtained among these evaluations is the absolute maximum, and the lowest value is the absolute minimum. Mathematically, if f(x) is the given function, find the critical points by solving f'(x) = 0 or f'(x) undefined. Evaluate f(x) at these critical points and the endpoints of the function's domain. The highest and lowest values among these evaluations represent the absolute maximum and minimum, respectively.
In summary, the process involves a combination of calculus and careful analysis of the function's domain. By finding critical points and evaluating the function at these points and the endpoints, you can identify the absolute maximum and minimum. This method is applicable to a wide range of functions, providing a systematic approach to locating extreme values.