a) Prove that for every e such that 0 < € < M/2 there is a non-empty open interval I≤ [a, b] such that cb (M - €)P|I| ≤ * \ƒ(x)\Pdx ≤ M²(b − a). b) Deduce that cb 1/p Him: (/* IF(x)/'da) ""- \ƒ(x)\³dx) = P→∞ = M. =
a) Prove that for every e such that 0 < € < M/2 there is a non-empty open interval I≤ [a, b] such that cb (M - €)P|I| ≤ * \ƒ(x)\Pdx ≤ M²(b − a). b) Deduce that cb 1/p Him: (/* IF(x)/'da) ""- \ƒ(x)\³dx) = P→∞ = M. =
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
![5. For the purposes of this exercise we will take for granted that continuous
functions on closed bounded interval are integrable (we shall prove it shortly).
supre[a,b]f(x)\. Suppose that
Let f : [a, b] → R be continuous and let M
M> 0. Let p > 0.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F008f8cce-1e45-43a4-8b17-46721d7357f5%2F80d67d18-35fe-4bd1-90e7-64ef75d2e356%2Fmummhzt_processed.png&w=3840&q=75)
Transcribed Image Text:5. For the purposes of this exercise we will take for granted that continuous
functions on closed bounded interval are integrable (we shall prove it shortly).
supre[a,b]f(x)\. Suppose that
Let f : [a, b] → R be continuous and let M
M> 0. Let p > 0.
=
![a) Prove that for every such that 0 < € < M/2 there is a non-empty
open interval IC [a, b] such that
(M — c)º|I| ≤ [* |ƒ(a)|³dx ≤ M²(b − a).
a
b) Deduce that
lim
P→∞
Cisco
|ƒ(x)|Pdx
1/p
= M.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F008f8cce-1e45-43a4-8b17-46721d7357f5%2F80d67d18-35fe-4bd1-90e7-64ef75d2e356%2Fyd0eet5_processed.png&w=3840&q=75)
Transcribed Image Text:a) Prove that for every such that 0 < € < M/2 there is a non-empty
open interval IC [a, b] such that
(M — c)º|I| ≤ [* |ƒ(a)|³dx ≤ M²(b − a).
a
b) Deduce that
lim
P→∞
Cisco
|ƒ(x)|Pdx
1/p
= M.
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