Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

검증된 단계별 안내

Step 1: Recall that the graph provided is of f'(x), the derivative of f(x). The derivative indicates the slope of the tangent line to the graph of f(x). When f'(x) > 0, f(x) is increasing, and when f'(x) < 0, f(x) is decreasing.

Step 2: Analyze the graph of f'(x). Identify the intervals where f'(x) is positive (above the x-axis) and where f'(x) is negative (below the x-axis). This will help determine the intervals of increase and decrease for f(x).

Step 3: From the graph, observe that f'(x) is positive on the interval (-2, -1) and (0, 1). This means f(x) is increasing on these intervals.

Step 4: Similarly, observe that f'(x) is negative on the interval (-1, 0) and (1, 2). This means f(x) is decreasing on these intervals.

Step 5: Summarize the findings: f(x) is increasing on (-2, -1) and (0, 1), and f(x) is decreasing on (-1, 0) and (1, 2). This information is determined directly from the graph of f'(x).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

First Derivative Test

The First Derivative Test is used to determine where a function is increasing or decreasing. If the derivative f'(x) is positive over an interval, the function f(x) is increasing on that interval. Conversely, if f'(x) is negative, f(x) is decreasing. This test helps identify local maxima and minima by analyzing the sign changes of f'(x).

Critical Points

Critical points occur where the derivative f'(x) is zero or undefined. These points are potential locations for local extrema (maxima or minima) of the function f(x). By examining the behavior of f'(x) around these points, one can determine the nature of the extrema using the First Derivative Test or the Second Derivative Test.

Graph Interpretation

Interpreting the graph of a derivative involves understanding how the slope of the tangent line to the function f(x) changes. The graph of f'(x) provides visual cues about the intervals of increase and decrease of f(x). Positive values of f'(x) indicate increasing intervals, while negative values indicate decreasing intervals, helping to identify the behavior of the original function.

Identifying ExtremaIn Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.