iii) Prove that if f is continuous, f(x) > 0 on [a, b], and f(x)dx = 0, then f is identically 0 on [a, b). Hint: Assume it is not. Then f(c) > 0 for some c E (a, b). Now use part ii). %3D
iii) Prove that if f is continuous, f(x) > 0 on [a, b], and f(x)dx = 0, then f is identically 0 on [a, b). Hint: Assume it is not. Then f(c) > 0 for some c E (a, b). Now use part ii). %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
#2. iii. 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbb66fa7-7c22-4982-a22f-aaed542f65b3%2F9659833d-a570-475a-b008-ff55befe02d5%2F9nysqjs_processed.png&w=3840&q=75)
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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