Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

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Examine the graph carefully to identify any critical points within the interval [a, b]. Critical points occur where the derivative is zero or undefined, which typically corresponds to peaks, troughs, or points of inflection on the graph.

Identify any endpoints of the interval [a, b] as potential candidates for absolute extreme values. Remember that absolute extrema can occur at endpoints or critical points within the interval.

Determine the local extreme values by looking for points where the graph changes direction. A local maximum occurs where the graph changes from increasing to decreasing, and a local minimum occurs where it changes from decreasing to increasing.

Evaluate the function at each critical point and endpoint to determine the function values. Compare these values to identify the absolute maximum and minimum values on the interval.

Summarize your findings by listing the points where local and absolute extreme values occur, specifying whether each is a maximum or minimum.

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주요 개념

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Local Extrema

Local extrema refer to points in a function where the function value is either a local maximum or minimum compared to nearby points. A local maximum occurs when the function value is higher than its immediate neighbors, while a local minimum occurs when it is lower. These points are critical for understanding the behavior of the function within a specific interval.

Absolute Extrema

Absolute extrema are the highest or lowest points of a function over a given interval, including endpoints. An absolute maximum is the largest value the function attains on that interval, while an absolute minimum is the smallest. Identifying these points is essential for determining the overall range of the function within the specified bounds.

Critical Points

Critical points are values in the domain of a function where the derivative is either zero or undefined. These points are significant because they are potential locations for local extrema. To find local and absolute extrema, one must evaluate the function at these critical points as well as at the endpoints of the interval.