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The graph of y = g(x) is shown below. At which point is g"< g'< g? A с D

The graph of y = g(x) is shown below. At which point is g"< g'< g? A с D

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Understanding Critical Points in Calculus In the study of calculus, it is crucial to understand the behavior of a function and its derivatives. Given the function y = g(x) as shown in the graph, you can determine several key properties about the function's rate of change and concavity by analyzing the function and its critical points. The graphed function y = g(x) presents us with several points of interest, denoted A, B, C, D, and E. **Problem:** At which point is \( g'' < g' < g \)? #### Graph Analysis: 1. **Points on the Graph:** - **A:** A point where the function appears to be increasing. - **B:** A local maximum where the derivative \( g'(x) \) is zero. - **C:** A point where the function is decreasing after the local maximum. - **D:** Likely a point close to the inflection point where the concavity changes. - **E:** A local minimum where again the derivative \( g'(x) \) is zero. 2. **Derivative and Concavity:** - **First Derivative \( g'(x) \):** Denotes the rate of change (slope) of the function. When positive, the function is increasing; when negative, it is decreasing. - **Second Derivative \( g''(x) \):** Denotes the concavity of the function. When positive, the graph is concave up; when negative, it is concave down. #### Points Analysis: - At **Point C**, the graph of the function is decreasing, indicating that \( g'(x) < 0 \). It is also concave down at this point, which means \( g''(x) < 0 \). Finally, since we are analyzing near a peak, \( g(x) \) still holds the greatest value among the first and second derivatives. Thus, **Point C** is where \( g'' < g' < g \) holds true. #### Conclusion The proper analysis of this function \( y = g(x) \) reveals that point C is the correct answer where \( g'' < g' < g \). Understanding such relationships between the function and its derivatives is fundamental in calculus, particularly for analyzing and interpreting the behavior of graphs. **Answer:** C
Transcribed Image Text:### Understanding Critical Points in Calculus In the study of calculus, it is crucial to understand the behavior of a function and its derivatives. Given the function y = g(x) as shown in the graph, you can determine several key properties about the function's rate of change and concavity by analyzing the function and its critical points. The graphed function y = g(x) presents us with several points of interest, denoted A, B, C, D, and E. **Problem:** At which point is \( g'' < g' < g \)? #### Graph Analysis: 1. **Points on the Graph:** - **A:** A point where the function appears to be increasing. - **B:** A local maximum where the derivative \( g'(x) \) is zero. - **C:** A point where the function is decreasing after the local maximum. - **D:** Likely a point close to the inflection point where the concavity changes. - **E:** A local minimum where again the derivative \( g'(x) \) is zero. 2. **Derivative and Concavity:** - **First Derivative \( g'(x) \):** Denotes the rate of change (slope) of the function. When positive, the function is increasing; when negative, it is decreasing. - **Second Derivative \( g''(x) \):** Denotes the concavity of the function. When positive, the graph is concave up; when negative, it is concave down. #### Points Analysis: - At **Point C**, the graph of the function is decreasing, indicating that \( g'(x) < 0 \). It is also concave down at this point, which means \( g''(x) < 0 \). Finally, since we are analyzing near a peak, \( g(x) \) still holds the greatest value among the first and second derivatives. Thus, **Point C** is where \( g'' < g' < g \) holds true. #### Conclusion The proper analysis of this function \( y = g(x) \) reveals that point C is the correct answer where \( g'' < g' < g \). Understanding such relationships between the function and its derivatives is fundamental in calculus, particularly for analyzing and interpreting the behavior of graphs. **Answer:** C
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