In Exercises 37-40 , locate the absolute extrema of the...

In Exercises $37-40$ , locate the absolute extrema of the function (if any exist) over each interval.
$$
\begin{array}{ll}{f(x)=2 x-3} \\ {\text { (a) }[0,2]} & {\text { (b) }[0,2)} \\ {\text { (c) }(0,2]} & {\text { (d) }(0,2)}\end{array}
$$

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Key Concepts

Absolute Extrema

Absolute extrema refer to the highest (maximum) and lowest (minimum) values that a function attains over a specified domain. In optimization problems, determining the absolute extrema involves evaluating the function at critical points and the endpoints of the interval, if they are included.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed interval, then it must have both an absolute maximum and an absolute minimum on that interval. This theorem is essential when deciding if a function is guaranteed to attain its extreme values.

Closed and Open Intervals

Understanding the distinction between closed and open intervals is crucial since the inclusion or exclusion of endpoints affects whether a function can attain its extrema at those endpoints. Closed intervals have endpoints included, ensuring that all boundary values are considered, while open intervals do not include one or both endpoints, which can lead to the absence of absolute extrema.

Monotonic Functions

Monotonic functions, which are either entirely non-increasing or non-decreasing over an interval, simplify the analysis of extrema since their absolute maximum or minimum often occurs at the endpoints of the interval. This property is particularly useful when dealing with linear functions.

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