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Use a second derivative test with a well written conclusion to find the inflections point of the following: distance using the seof(x) = √xe-*

Use a second derivative test with a well written conclusion to find the inflections point of the following: distance using the seof(x) = √xe-*

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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**Problem: Calculating Inflection Points Using the Second Derivative Test** Task: Use a second derivative test with a well-written conclusion to find the inflection points of the following function: \[ f(x) = \sqrt{x} e^{-x} \] To solve this problem, follow these steps: 1. **Differentiate the Function**: - Find the first derivative, \( f'(x) \). - Find the second derivative, \( f''(x) \). 2. **Find Critical Points**: - Determine where the second derivative is equal to zero or undefined. 3. **Second Derivative Test**: - Evaluate the sign of \( f''(x) \) around the critical points to determine the nature of these points (inflection points, local minima, or maxima). 4. **Conclusion**: - Provide a detailed explanation of the findings and interpret the results in the context of the problem.
Transcribed Image Text:**Problem: Calculating Inflection Points Using the Second Derivative Test** Task: Use a second derivative test with a well-written conclusion to find the inflection points of the following function: \[ f(x) = \sqrt{x} e^{-x} \] To solve this problem, follow these steps: 1. **Differentiate the Function**: - Find the first derivative, \( f'(x) \). - Find the second derivative, \( f''(x) \). 2. **Find Critical Points**: - Determine where the second derivative is equal to zero or undefined. 3. **Second Derivative Test**: - Evaluate the sign of \( f''(x) \) around the critical points to determine the nature of these points (inflection points, local minima, or maxima). 4. **Conclusion**: - Provide a detailed explanation of the findings and interpret the results in the context of the problem.
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