3.7 Suppose f and g are in R[a, b] and f(x) < g(x) for all r € [a, b]. Prove: f(x) dr < | g(x) dr. (Hint: Let h(x) = g(x) – f(x). Use Theorem 3.1.2 and Exercise 3.6 above.)
3.7 Suppose f and g are in R[a, b] and f(x) < g(x) for all r € [a, b]. Prove: f(x) dr < | g(x) dr. (Hint: Let h(x) = g(x) – f(x). Use Theorem 3.1.2 and Exercise 3.6 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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![3.7 Suppose f and g are in R[a, b] and f(x) < g(x) for all r € [a, b]. Prove:
| f(2) de <
g(z) dr.
(Hint: Let h(x) = g(x) – f(x). Use Theorem 3.1.2 and Exercise 3.6 above.)
For reference
3.6 † Suppose f E Rļa, b) and f(x) > 0 for all r E [a, b). Prove:
| f(x) dx > 0.
(Hint: Find a lower bound for all Riemann sums P(f, {7;}).)
Theorem 3.1.2 Let f and g be in R[a, b} and e e R. Then
i. f+9 € R[a, b] and
(f + g)(x) dx = | f(x) dx +
9(z) dz.
ii, ef E Rla, b) and
ef(r) dr = c
e f(z) der.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fadd5b56d-97e3-4d3c-9aa9-5b563ad77377%2F22ad1d18-f68f-4f49-8dc6-8f8da5e6a156%2F3z75abw_processed.png&w=3840&q=75)
Transcribed Image Text:3.7 Suppose f and g are in R[a, b] and f(x) < g(x) for all r € [a, b]. Prove:
| f(2) de <
g(z) dr.
(Hint: Let h(x) = g(x) – f(x). Use Theorem 3.1.2 and Exercise 3.6 above.)
For reference
3.6 † Suppose f E Rļa, b) and f(x) > 0 for all r E [a, b). Prove:
| f(x) dx > 0.
(Hint: Find a lower bound for all Riemann sums P(f, {7;}).)
Theorem 3.1.2 Let f and g be in R[a, b} and e e R. Then
i. f+9 € R[a, b] and
(f + g)(x) dx = | f(x) dx +
9(z) dz.
ii, ef E Rla, b) and
ef(r) dr = c
e f(z) der.
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