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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To determine where the function is increasing or decreasing, observe the graph of the function. The function is increasing where the slope of the graph is positive and decreasing where the slope is negative.

Identify the intervals where the function is increasing. From the graph, the function appears to be increasing on the interval (-2, 1).

Identify the intervals where the function is decreasing. From the graph, the function appears to be decreasing on the intervals (-∞, -2) and (1, ∞).

To find local extrema, look for points where the function changes from increasing to decreasing or vice versa. The graph shows a local maximum at x = -2 and a local minimum at x = 1.

Check for absolute extrema by comparing the values of the function at critical points and endpoints. The graph suggests an absolute maximum at x = -2 and an absolute minimum at x = 1.

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Increasing and Decreasing Intervals

A function is increasing on an interval if, for any two numbers x1 and x2 within that interval, x1 < x2 implies f(x1) < f(x2). Conversely, it is decreasing if x1 < x2 implies f(x1) > f(x2). Identifying these intervals involves analyzing the function's derivative: where it is positive, the function is increasing, and where it is negative, the function is decreasing.

Local and Absolute Extrema

Local extrema are points where a function reaches a minimum or maximum value within a neighborhood, while absolute extrema are the highest or lowest points over the entire domain. To find these, examine critical points where the derivative is zero or undefined, and evaluate the function's behavior at these points and endpoints of the domain.

Critical Points

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To determine the nature of these points, use the first or second derivative test. The first derivative test involves checking sign changes around the critical points, while the second derivative test uses concavity to identify extrema.

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.