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+ 12x - 3 - 3 =x, 0≤x≤ 2π sinx, 0≤x≤ 2π 12. f(x) = X x² + 1 2

+ 12x - 3 - 3 =x, 0≤x≤ 2π sinx, 0≤x≤ 2π 12. f(x) = X x² + 1 2

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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question 12
The image includes a calculus exercise with a graph and a list of problems related to functions and derivatives. Here is a detailed transcription and explanation: ### Problems 8–14: **Instructions:** - (a) Find the intervals on which \( f \) is increasing or decreasing. - (b) Find the local maximum and minimum values of \( f \). - (c) Find the intervals of concavity and the inflection points. **Functions:** 1. **9.** \( f(x) = x^3 - 3x^2 - 9x + 4 \) 2. **10.** \( f(x) = 2x^3 - 9x^2 + 12x - 3 \) 3. **11.** \( f(x) = x^4 - 2x^2 + 3 \) 4. **12.** \( f(x) = \frac{x}{x^2 + 1} \) 5. **13.** \( f(x) = \sin x + \cos x, \quad 0 \leq x \leq 2\pi \) 6. **14.** \( f(x) = \cos^2 x - 2 \sin x, \quad 0 \leq x \leq 2\pi \) ### Problems 15–17: **Instructions:** Find the local maximum and minimum values of \( f \) using both the First and Second Derivative Tests. Which method do you prefer? **Functions:** 1. **15.** \( f(x) = 1 + 3x^2 - 2x^3 \) 2. **16.** \( f(x) = \frac{x^2}{x - 1} \) 3. **17.** \( f(x) = \sqrt{x} - \sqrt[4]{x} \) ### Graph Explanation: The graph depicts the function \( y = f'(x) \), where the x-axis ranges from 0 to 9, and the y-axis contains the derivative values. The graph shows wave-like patterns indicating points where the function switches from increasing to decreasing, and vice versa. Points where the curve crosses the x-axis indicate possible local maxima, minima, and inflection points for \( f(x) \). ### Additional Instructions: Students are expected to apply knowledge of calculus to analyze the given functions
Transcribed Image Text:The image includes a calculus exercise with a graph and a list of problems related to functions and derivatives. Here is a detailed transcription and explanation: ### Problems 8–14: **Instructions:** - (a) Find the intervals on which \( f \) is increasing or decreasing. - (b) Find the local maximum and minimum values of \( f \). - (c) Find the intervals of concavity and the inflection points. **Functions:** 1. **9.** \( f(x) = x^3 - 3x^2 - 9x + 4 \) 2. **10.** \( f(x) = 2x^3 - 9x^2 + 12x - 3 \) 3. **11.** \( f(x) = x^4 - 2x^2 + 3 \) 4. **12.** \( f(x) = \frac{x}{x^2 + 1} \) 5. **13.** \( f(x) = \sin x + \cos x, \quad 0 \leq x \leq 2\pi \) 6. **14.** \( f(x) = \cos^2 x - 2 \sin x, \quad 0 \leq x \leq 2\pi \) ### Problems 15–17: **Instructions:** Find the local maximum and minimum values of \( f \) using both the First and Second Derivative Tests. Which method do you prefer? **Functions:** 1. **15.** \( f(x) = 1 + 3x^2 - 2x^3 \) 2. **16.** \( f(x) = \frac{x^2}{x - 1} \) 3. **17.** \( f(x) = \sqrt{x} - \sqrt[4]{x} \) ### Graph Explanation: The graph depicts the function \( y = f'(x) \), where the x-axis ranges from 0 to 9, and the y-axis contains the derivative values. The graph shows wave-like patterns indicating points where the function switches from increasing to decreasing, and vice versa. Points where the curve crosses the x-axis indicate possible local maxima, minima, and inflection points for \( f(x) \). ### Additional Instructions: Students are expected to apply knowledge of calculus to analyze the given functions
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