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Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(4) = 0, f'(x) = 1 if x < -1, f'(x) > 0 if 0 4, lim f'(x) = ∞, lim f'(x) = -∞, x→2+ f"(x) > 0 if – 1< x < 2 or 2 < x < 4, f"(x) < 0 if > 4

Sketch the graph of a function that satisfies all of the given conditions. f'(0) = f'(4) = 0, f'(x) = 1 if x < -1, f'(x) > 0 if 0 4, lim f'(x) = ∞, lim f'(x) = -∞, x→2+ f"(x) > 0 if – 1< x < 2 or 2 < x < 4, f"(x) < 0 if > 4

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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**Sketch the Graph of a Function with Given Conditions** Given Conditions: 1. \( f'(0) = f'(4) = 0 \) 2. \( f'(x) = 1 \) if \( x < -1 \) 3. \( f'(x) > 0 \) if \( 0 < x < 2 \) 4. \( f'(x) < 0 \) if \( -1 < x < 0 \) or \( 2 < x < 4 \) or \( x > 4 \) 5. \( \lim_{{x \to 2^-}} f'(x) = \infty \) 6. \( \lim_{{x \to 2^+}} f'(x) = -\infty \) 7. \( f''(x) > 0 \) if \( -1 < x < 2 \) or \( 2 < x < 4 \) 8. \( f''(x) < 0 \) if \( x > 4 \) **Explanation:** - \( f'(x) \) describes the slope or rate of change of the function. When \( f'(x) = 0 \), the function has a horizontal tangent, usually at local maxima, minima, or points of inflection. - \( f'(x) > 0 \) indicates the function is increasing. - \( f'(x) < 0 \) indicates the function is decreasing. - \( \lim_{{x \to 2^-}} f'(x) = \infty \) shows the slope approaches positive infinity as \( x \) approaches 2 from the left. - \( \lim_{{x \to 2^+}} f'(x) = -\infty \) shows the slope approaches negative infinity as \( x \) approaches 2 from the right, suggesting a vertical asymptote. - \( f''(x) \) is the second derivative, indicating concavity. \( f''(x) > 0 \) implies the function is concave up, and \( f''(x) < 0 \) means it is concave down.
Transcribed Image Text:**Sketch the Graph of a Function with Given Conditions** Given Conditions: 1. \( f'(0) = f'(4) = 0 \) 2. \( f'(x) = 1 \) if \( x < -1 \) 3. \( f'(x) > 0 \) if \( 0 < x < 2 \) 4. \( f'(x) < 0 \) if \( -1 < x < 0 \) or \( 2 < x < 4 \) or \( x > 4 \) 5. \( \lim_{{x \to 2^-}} f'(x) = \infty \) 6. \( \lim_{{x \to 2^+}} f'(x) = -\infty \) 7. \( f''(x) > 0 \) if \( -1 < x < 2 \) or \( 2 < x < 4 \) 8. \( f''(x) < 0 \) if \( x > 4 \) **Explanation:** - \( f'(x) \) describes the slope or rate of change of the function. When \( f'(x) = 0 \), the function has a horizontal tangent, usually at local maxima, minima, or points of inflection. - \( f'(x) > 0 \) indicates the function is increasing. - \( f'(x) < 0 \) indicates the function is decreasing. - \( \lim_{{x \to 2^-}} f'(x) = \infty \) shows the slope approaches positive infinity as \( x \) approaches 2 from the left. - \( \lim_{{x \to 2^+}} f'(x) = -\infty \) shows the slope approaches negative infinity as \( x \) approaches 2 from the right, suggesting a vertical asymptote. - \( f''(x) \) is the second derivative, indicating concavity. \( f''(x) > 0 \) implies the function is concave up, and \( f''(x) < 0 \) means it is concave down.
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