Sketch the graph of each function and determine whether the...

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

Open vs. Closed Intervals

The nature of the interval on which a function is defined significantly influences the existence of extreme values. Closed intervals, which include their endpoints, allow for the possibility of absolute extreme values due to the function being continuous over all limit points. In contrast, open intervals do not include their endpoints, and hence the function might approach but never attain certain values at the boundaries.

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 does not fully apply to functions defined on open intervals, where endpoints are not included, and as a result, some extreme values may not exist or be attained within that domain.

Graphing of Functions

Graphing functions involves plotting points that satisfy the function and connecting them to show the behavior of the function over its domain. In this context, the graph helps visualize how the function values change as x varies over the interval, highlighting potential maximum or minimum values.

Absolute Extreme Values

Absolute extreme values, or global extrema, refer to the highest (maximum) and lowest (minimum) values that a function attains on its domain. In analysis, determining whether these values exist on a given interval is essential, since they represent the bounds within which the function operates.

Transcript

Please give Ace some feedback