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 $1 .$
$$
y=\frac{6}{x^{2}+2}, \quad-1 < x <1
$$

Key Concepts

Extreme Value Theorem

The Extreme Value Theorem is a fundamental result in calculus that guarantees a continuous function on a closed and bounded interval will attain both an absolute maximum and an absolute minimum. This theorem is central when determining whether a function has extreme values and underpins the conditions necessary for their existence.

Continuity and Domain

Continuity ensures that there are no jumps or breaks in a function’s behavior across its domain, which is essential for the analysis of extreme values. However, even a continuous function may not have absolute extrema if its domain is not closed, emphasizing the importance of domain considerations alongside continuity.

Open vs. Closed Intervals

A closed interval includes its end points, while an open interval does not. The Extreme Value Theorem applies only to closed intervals, meaning that if a function is defined on an open interval, it might not attain its extreme values even though it approaches them, since the boundary points are not part of the domain.

Graphical Analysis and Boundary Behavior

Sketching the graph of a function aids in visualizing its behavior, especially in identifying local and absolute extrema. Analyzing how the function behaves near the endpoints of its domain, even if those endpoints are not included, is crucial for understanding the function's limits and whether true extreme values are achieved.

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