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Apply the second derivative test to find the absolute extrema of each function on (0, 0) 6. a) f(x) = 4 +x+- 8. b) f(x) = 2x+

Apply the second derivative test to find the absolute extrema of each function on (0, 0) 6. a) f(x) = 4 +x+- 8. b) f(x) = 2x+

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 2: Application of the Second Derivative Test #### Objective Apply the second derivative test to find the absolute extrema of each function on the interval \((0, \infty)\). #### Functions a) \( f(x) = 4 + x + \frac{9}{x} \) b) \( f(x) = 2x + \frac{8}{x} \) ##### Explanation For each function, the second derivative test involves the following steps: 1. **First Derivative (\(f'(x)\)):** - Calculate the first derivative of the function to find critical points. 2. **Critical Points:** - Set the first derivative equal to zero and solve for \(x\) to determine critical points. 3. **Second Derivative (\(f''(x)\)):** - Calculate the second derivative of the function. 4. **Apply the Second Derivative Test:** - Evaluate the second derivative at each critical point. - If \(f''(x) > 0\), the function has a local minimum at \(x\). - If \(f''(x) < 0\), the function has a local maximum at \(x\). - If \(f''(x) = 0\), the test is inconclusive. 5. **Identify Absolute Extrema:** - Compare the function values at critical points and endpoints (if applicable) to determine absolute extrema. Explore the detailed calculations and conclusions on the educational website to understand how the second derivative test identifies the nature of critical points and determines absolute extrema effectively.
Transcribed Image Text:### Problem 2: Application of the Second Derivative Test #### Objective Apply the second derivative test to find the absolute extrema of each function on the interval \((0, \infty)\). #### Functions a) \( f(x) = 4 + x + \frac{9}{x} \) b) \( f(x) = 2x + \frac{8}{x} \) ##### Explanation For each function, the second derivative test involves the following steps: 1. **First Derivative (\(f'(x)\)):** - Calculate the first derivative of the function to find critical points. 2. **Critical Points:** - Set the first derivative equal to zero and solve for \(x\) to determine critical points. 3. **Second Derivative (\(f''(x)\)):** - Calculate the second derivative of the function. 4. **Apply the Second Derivative Test:** - Evaluate the second derivative at each critical point. - If \(f''(x) > 0\), the function has a local minimum at \(x\). - If \(f''(x) < 0\), the function has a local maximum at \(x\). - If \(f''(x) = 0\), the test is inconclusive. 5. **Identify Absolute Extrema:** - Compare the function values at critical points and endpoints (if applicable) to determine absolute extrema. Explore the detailed calculations and conclusions on the educational website to understand how the second derivative test identifies the nature of critical points and determines absolute extrema effectively.
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