In Exercises 21-36, find the absolute maximum and minimum...

In Exercises $21-36,$ find the absolute maximum and minimum values
of each function on the given interval. Then graph the function. Identify the points
on the graph where the absolute extrema occur, and
include their coordinates.
$$
F(x)=-\frac{1}{x^{2}}, \quad 0.5 \leq x \leq 2
$$

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Key Concepts

Extreme Value Theorem

The Extreme Value Theorem guarantees that if a function is continuous on a closed interval, it will attain both an absolute maximum and an absolute minimum within that interval. This foundational concept underpins the process of finding extreme values by ensuring their existence when all conditions are met.

Critical Points

Critical points are values in the function's domain where its derivative is zero or does not exist. These points are essential in optimization problems because they are potential locations for local extreme values, and when analyzed along with the endpoints, they help determine the absolute maximum and minimum values on the interval.

Endpoint Evaluation

When dealing with a function on a closed interval, it is necessary to evaluate the function at the endpoints. This is because extreme values may occur at these boundaries rather than at interior critical points, making the examination of endpoints a crucial step in the overall strategy to find absolute extrema.

Graph Analysis

Graph analysis involves plotting the function to gain visual insights into its behavior, such as trends and turning points. This method aids in understanding the distribution of extreme values and provides a graphical verification of the analytical results obtained from evaluating critical points and endpoints.

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