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.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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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).
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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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