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3.22 If f e R[a, b], prove |f| € R[a, b]. (Hint: Use Exercises 3.20 and 3.21 above.)

3.22 If f e R[a, b], prove |f| € R[a, b]. (Hint: Use Exercises 3.20 and 3.21 above.)

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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3.22 If f € R[a, b), prove |f| € R[a, b]. (Hint: Use Exercises 3.20 and 3.21 above.) For reference 3.20 † Let f be any real-valued function on a domain DCR. Define f+(æ) = f(x) if f(x) > 0, if f(r) <0. f*(x) = and let -f(z) -f(r) if f(x) < 0, if f(x) 20 f (2) = for all r E D. Prove that f(x) = f*(x) – ƒ¯ (x) and |f(x)| = f+(x) + f¯(x) for all r e D. (Hint: Just check the cases based on the sign of f(r).) 3.21 1 Suppose f E R[a, b]. Prove: f+ and f- are in R[a, b). Hint: Show that U+, P) - L(f+, Р) <U(, P) — L(f, P).
Transcribed Image Text:3.22 If f € R[a, b), prove |f| € R[a, b]. (Hint: Use Exercises 3.20 and 3.21 above.) For reference 3.20 † Let f be any real-valued function on a domain DCR. Define f+(æ) = f(x) if f(x) > 0, if f(r) <0. f*(x) = and let -f(z) -f(r) if f(x) < 0, if f(x) 20 f (2) = for all r E D. Prove that f(x) = f*(x) – ƒ¯ (x) and |f(x)| = f+(x) + f¯(x) for all r e D. (Hint: Just check the cases based on the sign of f(r).) 3.21 1 Suppose f E R[a, b]. Prove: f+ and f- are in R[a, b). Hint: Show that U+, P) - L(f+, Р) <U(, P) — L(f, P).
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