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.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
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Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
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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.
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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