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Let ƒ(x) = ²⁄3 x³ – ½¼⁄x² + 5x + 3 be defined on the entire real line. Which of the following statements is true? Of has exactly one critical point located at x = 1. Of has one local minimum at x = 1 and another local minimum elsewhere. f has one local minimum at x = 1 and a local maximum elsewhere. ƒ has one local maximum at â = 1 and another local maximum elsewhere. has one local maximum at x = 1 and a local minimum elsewhere.

Let ƒ(x) = ²⁄3 x³ – ½¼⁄x² + 5x + 3 be defined on the entire real line. Which of the following statements is true? Of has exactly one critical point located at x = 1. Of has one local minimum at x = 1 and another local minimum elsewhere. f has one local minimum at x = 1 and a local maximum elsewhere. ƒ has one local maximum at â = 1 and another local maximum elsewhere. has one local maximum at x = 1 and a local minimum elsewhere.

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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The function \( f(x) = \frac{2}{3}x^3 - \frac{7}{2}x^2 + 5x + 3 \) is defined on the entire real line. We are asked to determine which of the following statements is true: 1. \( f \) has exactly one critical point located at \( x = 1 \). 2. \( f \) has one local minimum at \( x = 1 \) and another local minimum elsewhere. 3. \( f \) has one local minimum at \( x = 1 \) and a local maximum elsewhere. 4. \( f \) has one local maximum at \( x = 1 \) and another local maximum elsewhere. 5. \( f \) has one local maximum at \( x = 1 \) and a local minimum elsewhere. Definitions: - **Critical Point**: A point on the graph where the derivative is zero or undefined. - **Local Minimum/Maximum**: Points where the function attains a minimum/maximum value within a neighborhood. To determine the correct option, we can differentiate \( f(x) \) and investigate the critical points, as well as the nature of these points (whether they are local minima or maxima). The steps are: 1. Find the derivative \( f'(x) \). 2. Set \( f'(x) = 0 \) to find the critical points. 3. Use the second derivative test or examine changes in the sign of \( f'(x) \) to classify these critical points. Graph or diagram explanation: - If there were a graph of \( f(x) \) or its derivative, the critical points would be visible where the derivative \( f'(x) \) crosses the x-axis (indicating a zero), and the second derivative could indicate concavity (concave up for a minimum and concave down for a maximum). This problem requires calculus techniques to solve, specifically differentiation and critical point analysis.
Transcribed Image Text:The function \( f(x) = \frac{2}{3}x^3 - \frac{7}{2}x^2 + 5x + 3 \) is defined on the entire real line. We are asked to determine which of the following statements is true: 1. \( f \) has exactly one critical point located at \( x = 1 \). 2. \( f \) has one local minimum at \( x = 1 \) and another local minimum elsewhere. 3. \( f \) has one local minimum at \( x = 1 \) and a local maximum elsewhere. 4. \( f \) has one local maximum at \( x = 1 \) and another local maximum elsewhere. 5. \( f \) has one local maximum at \( x = 1 \) and a local minimum elsewhere. Definitions: - **Critical Point**: A point on the graph where the derivative is zero or undefined. - **Local Minimum/Maximum**: Points where the function attains a minimum/maximum value within a neighborhood. To determine the correct option, we can differentiate \( f(x) \) and investigate the critical points, as well as the nature of these points (whether they are local minima or maxima). The steps are: 1. Find the derivative \( f'(x) \). 2. Set \( f'(x) = 0 \) to find the critical points. 3. Use the second derivative test or examine changes in the sign of \( f'(x) \) to classify these critical points. Graph or diagram explanation: - If there were a graph of \( f(x) \) or its derivative, the critical points would be visible where the derivative \( f'(x) \) crosses the x-axis (indicating a zero), and the second derivative could indicate concavity (concave up for a minimum and concave down for a maximum). This problem requires calculus techniques to solve, specifically differentiation and critical point analysis.
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