Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits f'(a+) = lim f'(x), f'(b-)= lim f'(x) %3D both exist and are finite. Show that 1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x)= f(x) for all х€ (а, b). 2. If f'(a+) > m > f'(b–), then there exists c E (a, b) such that f'(c) = m.
Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits f'(a+) = lim f'(x), f'(b-)= lim f'(x) %3D both exist and are finite. Show that 1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x)= f(x) for all х€ (а, b). 2. If f'(a+) > m > f'(b–), then there exists c E (a, b) such that f'(c) = m.
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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![Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits
f'(a+) = lim f'(x),
f'(b-) = lim f'(æ)
T>a+
both exist and are finite. Show that
1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x) = f(x) for all
x E (a, b).
2. If f'(a+) > m > f'(b–), then there exists c E (a, b) such that f'(c) = m.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F400acf15-def3-488d-8f4f-c7c0fe39de4b%2F0ec62a17-4137-4580-9c45-99cec3d99659%2Faumscte_processed.png&w=3840&q=75)
Transcribed Image Text:Prove the following version of Darboux's Theorem: let f be differentiable in (a, b). Suppose that the two limits
f'(a+) = lim f'(x),
f'(b-) = lim f'(æ)
T>a+
both exist and are finite. Show that
1. (Existence of continuous extension) There is a function g(x) E C[a, b] such that g(x) = f(x) for all
x E (a, b).
2. If f'(a+) > m > f'(b–), then there exists c E (a, b) such that f'(c) = m.
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