Identifying Extrema

In Exercises 15–18:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

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Observe the graph of the function y = f(x) and identify the critical points where the function changes direction. These points are typically where the derivative is zero or undefined.

Determine the intervals where the function is increasing. A function is increasing on an interval if its derivative is positive over that interval. From the graph, identify the segments where the slope of the function is positive.

Determine the intervals where the function is decreasing. A function is decreasing on an interval if its derivative is negative over that interval. From the graph, identify the segments where the slope of the function is negative.

Identify the local extrema by examining the critical points. A local maximum occurs where the function changes from increasing to decreasing, and a local minimum occurs where the function changes from decreasing to increasing.

Identify the absolute extrema by comparing the values of the function at the critical points and endpoints of the interval. The highest value is the absolute maximum, and the lowest value is the absolute minimum.

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Increasing and Decreasing Intervals

A function is increasing on an interval if, for any two points within the interval, a larger x-value results in a larger y-value. Conversely, it is decreasing if a larger x-value results in a smaller y-value. To determine these intervals, one can analyze the graph's slope or use the first derivative test, where a positive derivative indicates increasing and a negative derivative indicates decreasing.

Local and Absolute Extrema

Local extrema are points where a function reaches a local maximum or minimum within a neighborhood, while absolute extrema are the highest or lowest points over the entire domain. Local extrema occur where the derivative changes sign, and absolute extrema can be found by evaluating the function at critical points and endpoints. The graph helps visualize these points as peaks (maxima) or troughs (minima).

Critical Points

Critical points of a function occur where its derivative is zero or undefined, indicating potential local maxima, minima, or points of inflection. These points are crucial for identifying intervals of increase or decrease and locating extrema. By analyzing the graph or calculating the derivative, one can pinpoint these critical points and assess their nature using the first or second derivative tests.

Identifying ExtremaIn Exercises 15–18:a. Find the open intervals on which the function is increasing and those on which it is decreasing.b. Identify the function’s local and absolute extreme values, if any, saying where they occur.