1. Sketch the graph of a function f with the following properties. f'(x) > 0, for x<0, f'(x) <0, for x > 0 ƒ" (x) > 0, for x < -1, f" (x) < 0, for -1 < x <1, f" (x) > 0, for x > 1 limx→∞ f (x) = limx→∞ f (x) = 0, f (0) 1 =
1. Sketch the graph of a function f with the following properties. f'(x) > 0, for x<0, f'(x) <0, for x > 0 ƒ" (x) > 0, for x < -1, f" (x) < 0, for -1 < x <1, f" (x) > 0, for x > 1 limx→∞ f (x) = limx→∞ f (x) = 0, f (0) 1 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. Sketch the graph of a function \( f \) with the following properties:
\[ f'(x) > 0, \text{ for } x < 0, \quad f'(x) < 0, \text{ for } x > 0 \]
\[ f''(x) > 0, \text{ for } x < -1, \quad f''(x) < 0, \text{ for } -1 < x < 1, \quad f''(x) > 0, \text{ for } x > 1 \]
\[ \lim_{{x \to -\infty}} f(x) = \lim_{{x \to \infty}} f(x) = 0, \quad f(0) = 1 \]
To explain the graphs and diagrams, consider these points:
- \( f'(x) \) represents the first derivative of \( f(x) \), implying the slope or the rate of change of the function \( f(x) \). For \( x < 0 \), the function is increasing since \( f'(x) > 0 \). For \( x > 0 \), the function is decreasing as \( f'(x) < 0 \).
- \( f''(x) \) is the second derivative of \( f(x) \), which indicates the concavity of the function. For \( x < -1 \), the graph is concave up as \( f''(x) > 0 \). For \( -1 < x < 1 \), the graph is concave down since \( f''(x) < 0 \). For \( x > 1 \), the graph is again concave up with \( f''(x) > 0 \).
- The limits \( \lim_{{x \to -\infty}} f(x) = 0 \) and \( \lim_{{x \to \infty}} f(x) = 0 \) suggest that as \( x \) approaches both negative and positive infinity, the function value approaches 0. This indicates horizontal asymptotes at \( y = 0 \).
- \( f(0) = 1 \) indicates that the function passes through the point (0,1).
Based on these properties, the function \( f(x) \) exhibits specific behaviors at different intervals of \( x \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52db8ee6-33d1-4aac-bbe4-ebc771de1e49%2F3d7c6c54-da30-4eb4-960d-2ab4d0b3345d%2Fz4z1xve_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Sketch the graph of a function \( f \) with the following properties:
\[ f'(x) > 0, \text{ for } x < 0, \quad f'(x) < 0, \text{ for } x > 0 \]
\[ f''(x) > 0, \text{ for } x < -1, \quad f''(x) < 0, \text{ for } -1 < x < 1, \quad f''(x) > 0, \text{ for } x > 1 \]
\[ \lim_{{x \to -\infty}} f(x) = \lim_{{x \to \infty}} f(x) = 0, \quad f(0) = 1 \]
To explain the graphs and diagrams, consider these points:
- \( f'(x) \) represents the first derivative of \( f(x) \), implying the slope or the rate of change of the function \( f(x) \). For \( x < 0 \), the function is increasing since \( f'(x) > 0 \). For \( x > 0 \), the function is decreasing as \( f'(x) < 0 \).
- \( f''(x) \) is the second derivative of \( f(x) \), which indicates the concavity of the function. For \( x < -1 \), the graph is concave up as \( f''(x) > 0 \). For \( -1 < x < 1 \), the graph is concave down since \( f''(x) < 0 \). For \( x > 1 \), the graph is again concave up with \( f''(x) > 0 \).
- The limits \( \lim_{{x \to -\infty}} f(x) = 0 \) and \( \lim_{{x \to \infty}} f(x) = 0 \) suggest that as \( x \) approaches both negative and positive infinity, the function value approaches 0. This indicates horizontal asymptotes at \( y = 0 \).
- \( f(0) = 1 \) indicates that the function passes through the point (0,1).
Based on these properties, the function \( f(x) \) exhibits specific behaviors at different intervals of \( x \
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