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Give an example of a nonzero continuous function f on [0,1] such that f f(x) dx = 0.

Give an example of a nonzero continuous function f on [0,1] such that f f(x) dx = 0.

Advanced Engineering Mathematics
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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#2 i. Thanks. 

2. i) Give an example of a nonzero continuous function f on [0, 1] such that f(r)dr = 0. ii) Prove that if f is continuous on [a, b], f(x) > 0 on [a, b], and if f(c) > 0 for some ce la, b], then i f(x)dx > 0. Hint: Use Exercise 4 iii from assignment 9 that says, if f is continuous and f(k) > 0, then we can find an interval (c, d) containing k such that f(x) > 0 on (c, d). Now use the additive property of R. I. , ſ, f(x)dx = , f(x)dx + Se f(x)dx+ S f (x)dx and the property that a positive function on an interval must have a positive integral. iii) Prove that if f is continuous, f(x) > 0 on [a, b], and f f(x)dx = 0, then f is identically O on [a, b]. Hint: Assume it is not. Then f(c) > 0 for some c E (a, b). Now use part i). %3D
Transcribed Image Text:2. i) Give an example of a nonzero continuous function f on [0, 1] such that f(r)dr = 0. ii) Prove that if f is continuous on [a, b], f(x) > 0 on [a, b], and if f(c) > 0 for some ce la, b], then i f(x)dx > 0. Hint: Use Exercise 4 iii from assignment 9 that says, if f is continuous and f(k) > 0, then we can find an interval (c, d) containing k such that f(x) > 0 on (c, d). Now use the additive property of R. I. , ſ, f(x)dx = , f(x)dx + Se f(x)dx+ S f (x)dx and the property that a positive function on an interval must have a positive integral. iii) Prove that if f is continuous, f(x) > 0 on [a, b], and f f(x)dx = 0, then f is identically O on [a, b]. Hint: Assume it is not. Then f(c) > 0 for some c E (a, b). Now use part i). %3D
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