4. (i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is CE (a, b) with f(c)(b – a) f(x) dx (ii) Assume that there exists cE (a, b) with 0. f(x) dx > f(x) dx < Show that there exists ro E (a, b) with f(xo): = 0. Hint: Apply the previous result to [a, c] and [c, b] and use the Intermediate Value Theorem.
4. (i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is CE (a, b) with f(c)(b – a) f(x) dx (ii) Assume that there exists cE (a, b) with 0. f(x) dx > f(x) dx < Show that there exists ro E (a, b) with f(xo): = 0. Hint: Apply the previous result to [a, c] and [c, b] and use the Intermediate Value Theorem.
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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Question
![(i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is
CE (a, b) with
4.
f(c)(b – a)
f(x) dx
(ii) Assume that there exists cE (a, b) with
| f(x) dx > 0,
f (x) dx < 0
Show that there exists xo E (a, b) with f(xo)
= 0. Hint: Apply the previous result to [a, c] and [c, b]
and use the Intermediate Value Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1db17ce4-1a18-4155-bd6b-6f94f4f69cf2%2Fabbf603b-5cb2-4e57-a1ae-b34e40aaf79b%2Fb590m0h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(i) Let f : [a, b] → R continuous. Use the Fundamental Theorem of Calculus to show that there is
CE (a, b) with
4.
f(c)(b – a)
f(x) dx
(ii) Assume that there exists cE (a, b) with
| f(x) dx > 0,
f (x) dx < 0
Show that there exists xo E (a, b) with f(xo)
= 0. Hint: Apply the previous result to [a, c] and [c, b]
and use the Intermediate Value Theorem.
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