For this problem, you will need to recall Definition 7.10 and Lemma 7.9. Assume that bounded functions f, g: [a, b] → R are integrable. (a) In this part you will show that if f(x) > 0 for all x € [a, b], then [ f(x) dx > 0 a by completing the following steps: (i) Show that L(f) ≥ 0. (ii) Use definition 7.10 to show that ·b [* a f(x) dx > 0.
For this problem, you will need to recall Definition 7.10 and Lemma 7.9. Assume that bounded functions f, g: [a, b] → R are integrable. (a) In this part you will show that if f(x) > 0 for all x € [a, b], then [ f(x) dx > 0 a by completing the following steps: (i) Show that L(f) ≥ 0. (ii) Use definition 7.10 to show that ·b [* a f(x) dx > 0.
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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Solve the following using defintions provided
![For this problem, you will need to recall Definition 7.10 and Lemma 7.9.
Assume that bounded functions f, g: [a, b] → R are integrable.
(a)
In this part you will show that if f(x) ≥ 0 for all x = [a, b], then
[
by completing the following steps:
(i) Show that L(f) ≥ 0.
(ii) Use definition 7.10 to show that
f(x) dx > 0
[ f(x) dx ≥ 0.
>](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecc418e3-a973-4d51-90e0-10d2ad078b59%2F98db3675-7849-4c38-ae47-91be55848f09%2Fymhhzw_processed.png&w=3840&q=75)
Transcribed Image Text:For this problem, you will need to recall Definition 7.10 and Lemma 7.9.
Assume that bounded functions f, g: [a, b] → R are integrable.
(a)
In this part you will show that if f(x) ≥ 0 for all x = [a, b], then
[
by completing the following steps:
(i) Show that L(f) ≥ 0.
(ii) Use definition 7.10 to show that
f(x) dx > 0
[ f(x) dx ≥ 0.
>
![Lemma 7.9) Let f [a.b] → R be a bounded function. If m ≤ f(x) < M Vx = [a,b],
then the following holds.
m(b − a) ≤ L(f) ≤ M(b − a)
Definition 7.10) A bounded function f : [a, b] → R is integrable if L(f) = U(f) and
is finite. Notation is as follows,
f* f(x)dx=L(f) = U (f)
a](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fecc418e3-a973-4d51-90e0-10d2ad078b59%2F98db3675-7849-4c38-ae47-91be55848f09%2F4qsp4gi_processed.png&w=3840&q=75)
Transcribed Image Text:Lemma 7.9) Let f [a.b] → R be a bounded function. If m ≤ f(x) < M Vx = [a,b],
then the following holds.
m(b − a) ≤ L(f) ≤ M(b − a)
Definition 7.10) A bounded function f : [a, b] → R is integrable if L(f) = U(f) and
is finite. Notation is as follows,
f* f(x)dx=L(f) = U (f)
a
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