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 $1 .$
$$
y=3 \sin x, \quad 0 < x < 2 \pi
$$

Key Concepts

Absolute Extreme Values

Absolute extreme values refer to the highest (maximum) and lowest (minimum) values that a function attains over a given domain. Determining these values involves analyzing the behavior of the function at critical points as well as at the boundaries of the domain, if they are included. This concept is fundamental in calculus and optimization problems, as it helps identify the most extreme outputs of a function.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it must attain both an absolute maximum and an absolute minimum on that interval. This theorem is crucial because it provides a guarantee for the existence of extreme values, but it only applies when the domain includes its endpoints, meaning the interval is closed.

Open vs Closed Intervals

The distinction between open and closed intervals is important when analyzing the extreme values of a function. In a closed interval, endpoints are included in the domain, which can allow for the attainment of extreme values at these boundary points. In contrast, an open interval does not include its endpoints, so even if a function approaches extreme values near these boundaries, it may not actually attain them. This subtle difference explains why certain continuous functions on open intervals might not have absolute extrema.

Graphical Analysis of Trigonometric Functions

Graphing trigonometric functions, such as sine functions, is an effective way to visualize their behavior over specific intervals. Graphical analysis can help identify where a function increases or decreases and where it reaches its peaks and valleys. This visual approach complements analytical methods, especially when considering the impact of domain restrictions on the existence of absolute extreme values.

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