In Exercises 15-20, sketch the graph of each function and...

In Exercises $15-20,$ sketch the graph of each function and determine
whether the function has any absolute extreme values on its domain.
Explain how your answer is consistent with Theorem $1 .$
$$
f(x)=|x|, \quad-1 < x < 2
$$

Key Concepts

Domain Restrictions and Endpoint Behavior

When analyzing a function’s extrema, understanding the domain is crucial. If the domain is an open interval, as in this problem, the endpoints are not included, which means that even though the function may approach certain values near the endpoints, it does not necessarily achieve them. This impacts the existence of absolute extrema and requires careful consideration when applying the Extreme Value Theorem.

Graphical Analysis of Functions

Graphing a function provides a visual insight into its behavior, including trends toward extreme values, continuity, and potential local or global extrema. For functions like the absolute value function, the graph reveals symmetry and a clear minimum point, while also illustrating how restricting the domain can affect the overall behavior and location of any absolute extrema.

Absolute Extremum

Absolute extrema refer to the highest (maximum) and lowest (minimum) values that a function attains on its domain. Identifying these values is central in optimization problems and calculus, though a function may not attain them within certain domains, particularly if the domain is not closed or if the extrema occur at non-included endpoints.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed, bounded interval, then it must attain both an absolute maximum and an absolute minimum on that interval. This theorem underpins many analyses in calculus by guaranteeing the existence of these extreme values under the specified conditions.

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