Final Answer:
The absolute minimum and maximum values are denoted by (a) Absolute Minimum and (b) Absolute Maximum, while (c) Relative Minimum and (d) Relative Maximum represent different types of local extrema in calculus.
Step-by-step explanation:
In calculus, determining the absolute maximum and minimum values of a function involves analyzing its critical points and endpoints. The absolute maximum and minimum are the highest and lowest points, respectively, that a function reaches over its entire domain.
Firstly, identify the critical points by finding where the derivative of the function is zero or undefined. Set the derivative equal to zero and solve for \(x\), then evaluate the function at these points. These critical values are potential locations for absolute maximum or minimum.
Next, consider the endpoints of the domain. Evaluate the function at the endpoints, and compare these values with the values obtained from the critical points. The highest among these values is the absolute maximum, and the lowest is the absolute minimum.
To summarize, finding the absolute maximum and minimum involves a systematic process of identifying critical points, evaluating the function at these points, and comparing these values with those at the endpoints. This approach ensures a comprehensive analysis of the function's behavior over its entire domain, leading to the determination of absolute maximum and minimum values.