In Exercises 37-40 , locate the absolute extrema of the...

In Exercises $37-40$ , locate the absolute extrema of the function (if any exist) over each interval.
$$
\begin{array}{l}{f(x)=\sqrt{4-x^{2}}} \\ {\text { (a) }[-2,2] \quad \text { (b) }[-2,0)} \\ {\text { (c) }(-2,2) \quad \text { (d) }[1,2)}\end{array}
$$

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest or lowest values that a function attains on a given set. In the context of calculus, determining the absolute maximum and minimum values on an interval involves evaluating both critical points where the derivative is zero or undefined and the endpoints of the interval (if they exist).

Critical Points

Critical points are points within the domain of the function where the derivative is zero or undefined. Identifying these points is essential for finding possible locations for local or absolute extrema, as these are the points where the function may change from increasing to decreasing or vice versa.

Extreme Value Theorem

The Extreme Value Theorem states that if a function is continuous on a closed, bounded interval, then it must have both an absolute maximum and an absolute minimum on that interval. This theorem plays a crucial role in determining the existence or non-existence of absolute extrema for continuous functions on closed intervals.

Open and Closed Intervals

Understanding the difference between open and closed intervals is important in the analysis of absolute extrema. A closed interval includes its endpoints, which must be evaluated when determining absolute extrema, while an open interval does not include its endpoints, potentially affecting the existence of absolute minimum or maximum values due to boundary behavior.

Domain of a Function

The domain of a function specifies the set of all input values for which the function is defined. In problems involving radicals or other functions with limited domains, it is critical to consider these domain restrictions when determining possible extrema, as values outside the domain cannot be included in the analysis.

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