Sketch a graph of a function f with the following properties.f' \u003c 0 and f" \u003c 0, for x \u003c -1

Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

문제 4.4.8a

4장, 문제 4.4.8a

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < -1

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Understand the given conditions: f'(x) < 0 implies the function f is decreasing, and f''(x) < 0 implies the function is concave down for x < -1.

For x < -1, since f'(x) < 0, the slope of the tangent line to the graph of f is negative, meaning the graph is sloping downward.

For x < -1, since f''(x) < 0, the graph is concave down, meaning it curves downward like an upside-down bowl.

Combine the two properties: The graph of f for x < -1 should be a decreasing curve that is concave down. This means the graph slopes downward and becomes steeper as x decreases.

Sketch the graph: Start at x = -1 and draw a curve that decreases and bends downward as x moves to the left. Ensure the curve reflects both the decreasing nature (f'(x) < 0) and the concave down property (f''(x) < 0).

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주요 개념

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

First Derivative Test

The first derivative of a function, denoted as f', indicates the slope of the tangent line to the graph of the function at any point. If f' < 0, the function is decreasing in that interval. Understanding this concept is crucial for sketching the graph, as it informs us that the function is moving downward for x < -1.

Second Derivative Test

The second derivative, denoted as f'', provides information about the concavity of the function. If f'' < 0, the function is concave down, meaning that the slope of the tangent line is decreasing. This concept is essential for understanding how the graph behaves in the specified interval, indicating that the function is not only decreasing but also bending downwards.

Graph Behavior

The overall behavior of a graph is influenced by both the first and second derivatives. In this case, since f' < 0 and f'' < 0 for x < -1, the graph will show a downward slope that becomes steeper as x decreases. Recognizing this behavior helps in accurately sketching the function, ensuring it reflects the properties of being decreasing and concave down.

Sketch a graph of a function f with the following properties.f' < 0 and f" < 0, for x < -1