Suppose ƒ : R → R is differentiable, f(0) = 0 and f'(x) > f(x) for all x ≥ 0. 1. Prove that f(x) > 0 on (0, a] for some a > 0. 2. Prove that f(x) > 0 for all x > 0.
Suppose ƒ : R → R is differentiable, f(0) = 0 and f'(x) > f(x) for all x ≥ 0. 1. Prove that f(x) > 0 on (0, a] for some a > 0. 2. Prove that f(x) > 0 for all x > 0.
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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![Suppose ƒ : R → R is differentiable, f(0) = 0 and ƒ'(x) > ƒ(x) for all x ≥ 0.
1. Prove that f(x) > 0 on (0, a] for some a > 0.
2. Prove that f(x) > 0 for all x > 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe15ed467-90ec-4e60-afef-3d3f6119f74d%2F3c59cb03-d2c1-415f-af41-53e6b2b9ea8d%2F2oxjdya_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose ƒ : R → R is differentiable, f(0) = 0 and ƒ'(x) > ƒ(x) for all x ≥ 0.
1. Prove that f(x) > 0 on (0, a] for some a > 0.
2. Prove that f(x) > 0 for all x > 0.
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