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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(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.)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.2: Integral Domains And Fields
Problem 24E: If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type...
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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.
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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