Please prove the Intermediate Value Theorem using the Fundamental Theorem of Calculus, part II and Darboux’s Theorem. All theorems are attached. Thank you.
Please prove the Intermediate Value Theorem using the Fundamental Theorem of Calculus, part II and Darboux’s Theorem. All theorems are attached. Thank you.
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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Please prove the Intermediate Value Theorem using the Fundamental Theorem of Calculus, part II and Darboux’s Theorem.
All theorems are attached. Thank you.
![Theorems:
1.
THEOREM 5.18 (Intermediate Value Theorem). If ƒ is continuous on [a, b] and a is any numbe
between f(a) and f(b), there there is some c e (a, b) for which f(c) = a.
2.
(Darboux's Theorem). Suppose f : [a, b] → R is differentiable. If a is between f'(a)
and f'(b), then there exists c E (a, b) where f'(c) = a.
3.
F: (a, b]
(The Fundamental Theorem of Calculus, part II). If ƒ : [a, b] → R is integrable and
→ R satisfies F" (x) = f(x), then
| f(2)dx = F(b) – F(a).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2f6386a-3cc2-426b-9dbe-808d2ada845c%2Faac852bc-0b28-4f7b-aefd-7698cd7101bb%2Fcbwfmt7_processed.png&w=3840&q=75)
Transcribed Image Text:Theorems:
1.
THEOREM 5.18 (Intermediate Value Theorem). If ƒ is continuous on [a, b] and a is any numbe
between f(a) and f(b), there there is some c e (a, b) for which f(c) = a.
2.
(Darboux's Theorem). Suppose f : [a, b] → R is differentiable. If a is between f'(a)
and f'(b), then there exists c E (a, b) where f'(c) = a.
3.
F: (a, b]
(The Fundamental Theorem of Calculus, part II). If ƒ : [a, b] → R is integrable and
→ R satisfies F" (x) = f(x), then
| f(2)dx = F(b) – F(a).
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