Hint: This follows easily from a certain string of inequalities; E 1-14 is relevant. *39. Suppose ƒ and g are integrable on [a, b] and g(x) 2 0 for all x is Let P be a partition of [a, b]. Let M{ and m,' denote the appi sup’s and inf's for f, define M" and m;" similarly for g, and de and m¡ similarly for fg.
Hint: This follows easily from a certain string of inequalities; E 1-14 is relevant. *39. Suppose ƒ and g are integrable on [a, b] and g(x) 2 0 for all x is Let P be a partition of [a, b]. Let M{ and m,' denote the appi sup’s and inf's for f, define M" and m;" similarly for g, and de and m¡ similarly for fg.
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 38 completely
![LTE
8:02 AM Ø
VOLTE L C 4 17%
*33.
(a) Suppose that f is continuous on [a, b] and l 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 = 0 for those con-
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.
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 |f].
(a) Prove that M,' – m²' < M¡ – mị.
(b) Prove that if ƒ 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 partť" min(f, 0) are integrable on [a, b].
38.
Prove that if f is integrable on
[a, b], then
Hint: This follows easily from a certain string of inequalities; Problem
1-14 is relevant.
*39. Suppose f and g are integrable on [a, b] and g(x) 2 0 for all x in [a, b].
Let P be a partition of [a, b]. Let M, and m' denote the appropriate
sup's and inf's for f, define M;" and m;" similarly for g, and define M;
and m¡ similarly for fg.
(a) Prove that M; < M¿'M;" and m; 2 m;'m;".
(b) Show that
U(fg, P) – L(fg, P) < ) [M¿'M;" – m{m;"](t; – i-1).
13. Integrals 26.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F49e60f1d-a03d-4abd-b02e-d7681ad9c98e%2F8ea2752a-9b8a-4d4b-ad86-cfe9864f2872%2Fu0l74oj_processed.png&w=3840&q=75)
Transcribed Image Text:LTE
8:02 AM Ø
VOLTE L C 4 17%
*33.
(a) Suppose that f is continuous on [a, b] and l 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 = 0 for those con-
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.
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 |f].
(a) Prove that M,' – m²' < M¡ – mị.
(b) Prove that if ƒ 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 partť" min(f, 0) are integrable on [a, b].
38.
Prove that if f is integrable on
[a, b], then
Hint: This follows easily from a certain string of inequalities; Problem
1-14 is relevant.
*39. Suppose f and g are integrable on [a, b] and g(x) 2 0 for all x in [a, b].
Let P be a partition of [a, b]. Let M, and m' denote the appropriate
sup's and inf's for f, define M;" and m;" similarly for g, and define M;
and m¡ similarly for fg.
(a) Prove that M; < M¿'M;" and m; 2 m;'m;".
(b) Show that
U(fg, P) – L(fg, P) < ) [M¿'M;" – m{m;"](t; – i-1).
13. Integrals 26.
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