Let f be a bounded function on [a, b] and let P be a partition of [a, b]. Let M; and m; have their usual meanings, and let M, and m{' have the corresponding meanings for the function |f|. (a) Prove that M{ – m' < M; - mị. (b) Prove that if f is integrable on [a, b], then so is |fl. (c) Prove that if ƒ and g are integrable on [a, b], then so are max(f, g) and min(f, g).
Let f be a bounded function on [a, b] and let P be a partition of [a, b]. Let M; and m; have their usual meanings, and let M, and m{' have the corresponding meanings for the function |f|. (a) Prove that M{ – m' < M; - mị. (b) Prove that if f is integrable on [a, b], then so is |fl. (c) Prove that if ƒ and g are integrable on [a, b], then so are max(f, g) and min(f, g).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve question 37 completely
![8:01 AM O
VOLTE ile 4 17%
*32.
Recall, from Problem 14, that f 2 0 if f(x) > 0 for all x in [a, b].
(a) Give an example where f(x) > 0 for all x, and f(x) > 0 for some x in
If = 0.
[а, b), and
(b) Suppose f(x) 2 0 for all x in [a, b] and f is continuous at xo in [a, b]
and f(xo) > 0. Prove that °f > 0. Hint: It suffices to find one
lower sum L(f, P) which is positive.
(c) Suppose f is integrable on [a, b] and f(x) > 0 for all x in [a, b].
Prove that "ƒ > 0. Hint: You will need Problem 31 ; indeed that
was one reason for including Problem 31.
(a) Suppose that f is continuous on [a, b] and fg = 0 for all con-
tinuous functions g on [a, b]. Prove that f = 0. (This is easy; there
is an obvious g to choose.)
(b) Suppose f is continuous on [a, b] and that /° fg
tinuous functions g on [a, b] which satisfy the extra conditions
g(a) = g(b) = 0. Prove that ƒ = 0. (This innocent looking fact is
an important lemma in the calculus of variations; see the Suggested
Reading for references.) Hint: Derive a contradiction from the
assumption f(xo) > 0 or f(xo) < 0; the g you pick will depend on
the behavior of f near xo.
*33.
= 0 for those con-
34. Let f(x) = x for x rational and f(x) = 0 for x irrational.
(a) Compute L(f, P) for all partitions P of [0, 1].
(b) Find inf {U(f, P): P a partition of [0, 1]}.
*35. Let f(x) = 0 for irrational x, and 1/q if x = p/q in lowest terms. Show
that f is integrable on [0, 1] and that f = 0. (Every lower sum is
clearly 0; you must figure out how to make upper sums small.)
*36. Find two functions f and g which are integrable, but whose composition
gof is not. Hint: Problem 35 is relevant.
*37. Let f be a bounded function on [a, b] and let P be a partition of [a, b].
Let M; and m; have their usual meanings, and let M,' and m' have the
corresponding meanings for the function fl.
(а) Prove that M; — m, < M; — т;.
(b) Prove that if f is integrable on [a, b], then so is |f].
(c) Prove that if f and g are integrable on [a, b], then so are max(f, g)
and min(f, g).
(d) Prove that f is integrable on [a, b] if and only if its "positive part"
max(f, 0) and its “negative part" min(f, 0) are integrable on [a, b].
38.
Prove that if f is integrable on [a, b], then
Hint: Thie fellewa](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49e60f1d-a03d-4abd-b02e-d7681ad9c98e%2Fcb117c50-e1ed-4d9a-8165-cbfef00ffdb6%2Frqskpnd_processed.png&w=3840&q=75)
Transcribed Image Text:8:01 AM O
VOLTE ile 4 17%
*32.
Recall, from Problem 14, that f 2 0 if f(x) > 0 for all x in [a, b].
(a) Give an example where f(x) > 0 for all x, and f(x) > 0 for some x in
If = 0.
[а, b), and
(b) Suppose f(x) 2 0 for all x in [a, b] and f is continuous at xo in [a, b]
and f(xo) > 0. Prove that °f > 0. Hint: It suffices to find one
lower sum L(f, P) which is positive.
(c) Suppose f is integrable on [a, b] and f(x) > 0 for all x in [a, b].
Prove that "ƒ > 0. Hint: You will need Problem 31 ; indeed that
was one reason for including Problem 31.
(a) Suppose that f is continuous on [a, b] and fg = 0 for all con-
tinuous functions g on [a, b]. Prove that f = 0. (This is easy; there
is an obvious g to choose.)
(b) Suppose f is continuous on [a, b] and that /° fg
tinuous functions g on [a, b] which satisfy the extra conditions
g(a) = g(b) = 0. Prove that ƒ = 0. (This innocent looking fact is
an important lemma in the calculus of variations; see the Suggested
Reading for references.) Hint: Derive a contradiction from the
assumption f(xo) > 0 or f(xo) < 0; the g you pick will depend on
the behavior of f near xo.
*33.
= 0 for those con-
34. Let f(x) = x for x rational and f(x) = 0 for x irrational.
(a) Compute L(f, P) for all partitions P of [0, 1].
(b) Find inf {U(f, P): P a partition of [0, 1]}.
*35. Let f(x) = 0 for irrational x, and 1/q if x = p/q in lowest terms. Show
that f is integrable on [0, 1] and that f = 0. (Every lower sum is
clearly 0; you must figure out how to make upper sums small.)
*36. Find two functions f and g which are integrable, but whose composition
gof is not. Hint: Problem 35 is relevant.
*37. Let f be a bounded function on [a, b] and let P be a partition of [a, b].
Let M; and m; have their usual meanings, and let M,' and m' have the
corresponding meanings for the function fl.
(а) Prove that M; — m, < M; — т;.
(b) Prove that if f is integrable on [a, b], then so is |f].
(c) Prove that if f and g are integrable on [a, b], then so are max(f, g)
and min(f, g).
(d) Prove that f is integrable on [a, b] if and only if its "positive part"
max(f, 0) and its “negative part" min(f, 0) are integrable on [a, b].
38.
Prove that if f is integrable on [a, b], then
Hint: Thie fellewa
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