7. Let f, g: [0, 0) → R be continuous on [0, ∞) and differentiable on (0, o0). If f(0) = g(0) and f'(r) > g(r) for all r E (0, o0), prove that f(r) > g(x) for all IE [0, 00).
7. Let f, g: [0, 0) → R be continuous on [0, ∞) and differentiable on (0, o0). If f(0) = g(0) and f'(r) > g(r) for all r E (0, o0), prove that f(r) > g(x) for all IE [0, 00).
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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Number 7 please
![7. Let f, g : [0, o) → R be continuous on [0, 00) and differentiable on (0, 00). If f(0) = g(0) and f'(z) > g(r)
for all r E (0, ), prove that f(x) > g(x) for all r E [0, 00).
8. A differentiable function f: [a,b] → R is uniformly differentiable on [a, b] if for every e > 0, there exists a
f'(r) < e. Prove that f is uniformly
f(t) - f(r)
8 > 0 such that for all t, r € [a, b] with 0 < |t – r| < 6, (0) = 7(4) -
8 >0 such that for all t, r E [a, b] with 0 < |t - r < 6,
t- x
differentiable on [a, b] if and only if f' is continuous on a, b.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F17d5e9b1-ab49-4794-ac8d-2bc7d1c0ce5b%2F19517de0-1005-47ca-9734-e7969bfa7dcb%2F4rf93r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. Let f, g : [0, o) → R be continuous on [0, 00) and differentiable on (0, 00). If f(0) = g(0) and f'(z) > g(r)
for all r E (0, ), prove that f(x) > g(x) for all r E [0, 00).
8. A differentiable function f: [a,b] → R is uniformly differentiable on [a, b] if for every e > 0, there exists a
f'(r) < e. Prove that f is uniformly
f(t) - f(r)
8 > 0 such that for all t, r € [a, b] with 0 < |t – r| < 6, (0) = 7(4) -
8 >0 such that for all t, r E [a, b] with 0 < |t - r < 6,
t- x
differentiable on [a, b] if and only if f' is continuous on a, b.
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