Recall that a function : RR satisfies (called a Lipschitz constant) such that a Lipschitz condition on an interval [a, b] if there exists a constant L |f(u)-f(v)| ≤ Lu-v| for all u, v € [a, b]. For each of the following cases, either derive a suitable Lipschitz constant L, with justification, or prove that no suitable Lipschitz constant exists: a f(x) = ex on the interval [-1, 1]. b f(x) = ex on the interval R. c f(x) = sign(x) on the interval [-1, 1], where the function sign is defined as sign(x) = { 1 0 -1 if x > 0, if x = 0, if x < 0.
Recall that a function : RR satisfies (called a Lipschitz constant) such that a Lipschitz condition on an interval [a, b] if there exists a constant L |f(u)-f(v)| ≤ Lu-v| for all u, v € [a, b]. For each of the following cases, either derive a suitable Lipschitz constant L, with justification, or prove that no suitable Lipschitz constant exists: a f(x) = ex on the interval [-1, 1]. b f(x) = ex on the interval R. c f(x) = sign(x) on the interval [-1, 1], where the function sign is defined as sign(x) = { 1 0 -1 if x > 0, if x = 0, if x < 0.
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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Question
![P1 Recall that a function f : R → R satisfies
(called a Lipschitz constant) such that
a Lipschitz condition on an interval [a, b] if there exists a constant L
|f(u) = f(v)| ≤ Lu-v|
for all u, v € [a, b]. For each of the following cases, either derive a suitable Lipschitz constant L, with justification,
or prove that no suitable Lipschitz constant exists:
a f(x) = ex on the interval [-1,1].
b f(x) = ex on the interval R.
c f(x) = sign(x) on the interval [-1, 1], where the function sign is defined as
sign(x) =
{
1
0
-1
if x > 0,
if x = 0,
if x < 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F008f8cce-1e45-43a4-8b17-46721d7357f5%2Fcef4fb8a-99f7-4928-a929-e5b02b606443%2F24mdp65_processed.jpeg&w=3840&q=75)
Transcribed Image Text:P1 Recall that a function f : R → R satisfies
(called a Lipschitz constant) such that
a Lipschitz condition on an interval [a, b] if there exists a constant L
|f(u) = f(v)| ≤ Lu-v|
for all u, v € [a, b]. For each of the following cases, either derive a suitable Lipschitz constant L, with justification,
or prove that no suitable Lipschitz constant exists:
a f(x) = ex on the interval [-1,1].
b f(x) = ex on the interval R.
c f(x) = sign(x) on the interval [-1, 1], where the function sign is defined as
sign(x) =
{
1
0
-1
if x > 0,
if x = 0,
if x < 0.
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