Find the absolute maximum and minimum values of each...

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$$f(x)=\frac{2}{3} x-5, \quad-2 \leq x \leq 3$$

Key Concepts

Graphical Representation and Coordinate Identification

Graphing a function not only visualizes its behavior over a given interval but also helps in identifying key features such as the absolute maximum and minimum. By plotting the function and marking the coordinates of these extreme points, one can clearly see and communicate where the function reaches its highest and lowest values, which is especially useful in applications and further analysis.

Linear Function Properties

Linear functions, typically expressed in the form f(x) = mx + b, have a constant rate of change. Their monotonic nature—either strictly increasing or decreasing—means that on any closed interval, the absolute maximum and minimum values will occur at the endpoints of the interval, eliminating the need to check for interior critical points as one would with non-linear functions.

Extreme Value Theorem

The Extreme Value Theorem is a fundamental concept in calculus which states that if a function is continuous on a closed interval, then it must attain both an absolute maximum and an absolute minimum value on that interval. This theorem guarantees that when looking for extreme values, one will always find them provided the continuity and closed interval conditions are met.

Endpoint Evaluation

When determining absolute extrema on a closed interval, it is essential to evaluate the function at its endpoints. Since the extreme values can occur either at points where the derivative is zero (critical points) or at the endpoints of the interval, checking the endpoints ensures that all candidates for maximum and minimum values are considered.

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