In the above space, draw the graph of a function with ALL of the following properties: (a) f(0) = -2, (b) lim f(x) = 4, (c) lim f(x) = -4, (d) f(x) has a vertical asymptote at x =-1, (e) f'(x) > 0 for -o 0 for -o < x < -5 and -1 < x < 3, (h) f"(x) < 0 for -5 < x < -1 and 3 < x
In the above space, draw the graph of a function with ALL of the following properties: (a) f(0) = -2, (b) lim f(x) = 4, (c) lim f(x) = -4, (d) f(x) has a vertical asymptote at x =-1, (e) f'(x) > 0 for -o 0 for -o < x < -5 and -1 < x < 3, (h) f"(x) < 0 for -5 < x < -1 and 3 < x
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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Question
Transcribed Image Text:In this exercise, we aim to draw the graph of a function that satisfies all of the following properties:
(a) \( f(0) = -2 \),
(b) \( \lim_{x \to \infty} f(x) = 4 \),
(c) \( \lim_{x \to -\infty} f(x) = -4 \),
(d) \( f(x) \) has a vertical asymptote at \( x = -1 \),
(e) \( f'(x) > 0 \) for \( -\infty < x < -3 \) and \( 1 < x < \infty \),
(f) \( f'(x) < 0 \) for \( -3 < x < -1 \) and \( -1 < x < 1 \),
(g) \( f''(x) > 0 \) for \( -\infty < x < -5 \) and \( -1 < x < 3 \),
(h) \( f''(x) < 0 \) for \( -5 < x < -1 \) and \( 3 < x < \infty \).
These properties describe specific behaviors of the function, including its intercept, limits, asymptotic behavior, intervals of increase and decrease, and concavity.
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