Exercise 4. (i) Suppose that u :]0, 1] → R is a bounded function. Define a function v: [0, 1] → R by v(0) = 0 and, for all x €]0, 1], v(x) = x²u(x). Show that v is differentiable in 0. (ii) Find a function v: [0, 1] → R such that v is differentiable on [0, 1] and v' is unbounded on [0, 1]. You may want to apply the preceding construction to a rapidly oscillating function u.
Exercise 4. (i) Suppose that u :]0, 1] → R is a bounded function. Define a function v: [0, 1] → R by v(0) = 0 and, for all x €]0, 1], v(x) = x²u(x). Show that v is differentiable in 0. (ii) Find a function v: [0, 1] → R such that v is differentiable on [0, 1] and v' is unbounded on [0, 1]. You may want to apply the preceding construction to a rapidly oscillating function u.
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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![Exercise 4. (i) Suppose that u :]0, 1] → R is a bounded function. Define
a function v: [0, 1] → R by v(0) = 0 and, for all x ]0, 1], v(x) = x²u(x).
Show that v is differentiable in 0.
(ii) Find a function v: [0, 1] → R such that v is differentiable on [0, 1] and v'
is unbounded on [0, 1]. You may want to apply the preceding construction
to a rapidly oscillating function u.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F702b72a5-f07f-4c22-a3f5-6956511ed203%2F14c37bdd-0dae-4f99-aa36-e4b2ebd55898%2F8r6g0ul_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 4. (i) Suppose that u :]0, 1] → R is a bounded function. Define
a function v: [0, 1] → R by v(0) = 0 and, for all x ]0, 1], v(x) = x²u(x).
Show that v is differentiable in 0.
(ii) Find a function v: [0, 1] → R such that v is differentiable on [0, 1] and v'
is unbounded on [0, 1]. You may want to apply the preceding construction
to a rapidly oscillating function u.
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