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In this part you will show that if f(x) ≤ g(x) for all x = [a, b], then [* f(x) dx ≤ [*9(x) dx by completing the following steps: First show that gf is a non-negative function on [a, b], that is, g-f≥ 0 on [a, b]. Using HW5, Exercise 8b prove that -f is integrable. Using HW5, Exercise 8c, prove that g - f is integrable on [a, b].

In this part you will show that if f(x) ≤ g(x) for all x = [a, b], then [* f(x) dx ≤ [*9(x) dx by completing the following steps: First show that gf is a non-negative function on [a, b], that is, g-f≥ 0 on [a, b]. Using HW5, Exercise 8b prove that -f is integrable. Using HW5, Exercise 8c, prove that g - f is integrable on [a, b].

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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Prove the following parts using everything provided

In this part you will show that if f(x) ≤ g(x) for all x € [a, b], then [1(x) dx < [*9(2) dz g(x) dx by completing the following steps: ●First show that g - f is a non-negative function on [a, b], that is, [a, b]. Using HW5, Exercise 8b prove that -f is integrable. Using HW5, Exercise 8c, prove that g - f is integrable on [a, b]. g-f≥ 0 on
Transcribed Image Text:In this part you will show that if f(x) ≤ g(x) for all x € [a, b], then [1(x) dx < [*9(2) dz g(x) dx by completing the following steps: ●First show that g - f is a non-negative function on [a, b], that is, [a, b]. Using HW5, Exercise 8b prove that -f is integrable. Using HW5, Exercise 8c, prove that g - f is integrable on [a, b]. g-f≥ 0 on
Lemma 7.9) Let f [ab] →→ R be a bounded function. If m ≤ f(x) < M Vr = [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(ƒ) = U(f) and is finite. Notation is as follows, HW5 ob f* f(x)dx=L(f) = U(ƒ) HVE (b) For any k € R, kf is also integrable on [a, b], and [k k. f(x) dx = k· (c) The function f+g is also integrable on [a, b], and [^(f(x) + a a f(x)dr + 9(x))dx = [* f(x)dx + - [* f(x)dx + [*9(x)dx a
Transcribed Image Text:Lemma 7.9) Let f [ab] →→ R be a bounded function. If m ≤ f(x) < M Vr = [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(ƒ) = U(f) and is finite. Notation is as follows, HW5 ob f* f(x)dx=L(f) = U(ƒ) HVE (b) For any k € R, kf is also integrable on [a, b], and [k k. f(x) dx = k· (c) The function f+g is also integrable on [a, b], and [^(f(x) + a a f(x)dr + 9(x))dx = [* f(x)dx + - [* f(x)dx + [*9(x)dx a
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