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
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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadd5b56d-97e3-4d3c-9aa9-5b563ad77377%2F8eeed78a-2e58-4488-aa7b-31f55ef4f946%2Fkw79cis_processed.png&w=3840&q=75)
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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