ii) Prove that if f is continuous on [a, b], ƒ(x) > 0 on [a, b], and if f(c) > 0 for some ce (a, b), then 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. , Sª f(x)dx = L% f(x)dx + Se f(x)dx+ [% f (x)dx and the property that a positive function on an interval must have a positive integral.
ii) Prove that if f is continuous on [a, b], ƒ(x) > 0 on [a, b], and if f(c) > 0 for some ce (a, b), then 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. , Sª f(x)dx = L% f(x)dx + Se f(x)dx+ [% f (x)dx and the property that a positive function on an interval must have a positive integral.
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 98E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
Related questions
Question
#2. ii. 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%2F289505ff-0e17-4ace-b475-446e8f6d788c%2Faqjvpbn_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
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781938168383/9781938168383_smallCoverImage.gif)
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781938168383/9781938168383_smallCoverImage.gif)
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781305115545/9781305115545_smallCoverImage.gif)
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning