Suppose a e R', fis a twice-differentiable real function on (a, ), and Mo, M1, M2 are the least upper bounds of |f(x)|, \f'(x)|, |f"(x)|, respectively, on (a, 0). Prove that M{ <4M, M2. Hint: If h >0, Taylor's theorem shows that f'(x) 2h [f(x+ 2h) – f(x)1– hf "() | for some g e (x, x + 2h). Hence Mo |f'(x)| SHM2 + h
Suppose a e R', fis a twice-differentiable real function on (a, ), and Mo, M1, M2 are the least upper bounds of |f(x)|, \f'(x)|, |f"(x)|, respectively, on (a, 0). Prove that M{ <4M, M2. Hint: If h >0, Taylor's theorem shows that f'(x) 2h [f(x+ 2h) – f(x)1– hf "() | for some g e (x, x + 2h). Hence Mo |f'(x)| SHM2 + h
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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![15. Suppose a e R', fis a twice-differentiable real function on (a, o), and Mo, M1, M2
are the least upper bounds of |f(x)|, |f"(x)|, [f"(x)|, respectively, on (a, 0).
Prove that
M{<4M, M2.
Hint: If h >0, Taylor's theorem shows that
f'(x) =
1
[f(x + 2h) – f(x)] – hf"()
2h
for some { e (x, x + 2h). Hence
Мо
If"(x)| <hM2 +
h
To show that M{ =4M, M2 can actually happen, take a = -1, define
(2x² – 1
(-1 < x < 0),
f(x) = {x? – 1
(0<x< 0),
(x² + 1
and show that Mo =1, M, =4, M2=4.
Does Mi <4M, M2 hold for vector-valued functions too?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38555b51-f7bb-437a-a331-1d6dd7be3349%2Fec4d75b6-8a2d-435a-bab0-83de3810021d%2F2mqkiaq_processed.png&w=3840&q=75)
Transcribed Image Text:15. Suppose a e R', fis a twice-differentiable real function on (a, o), and Mo, M1, M2
are the least upper bounds of |f(x)|, |f"(x)|, [f"(x)|, respectively, on (a, 0).
Prove that
M{<4M, M2.
Hint: If h >0, Taylor's theorem shows that
f'(x) =
1
[f(x + 2h) – f(x)] – hf"()
2h
for some { e (x, x + 2h). Hence
Мо
If"(x)| <hM2 +
h
To show that M{ =4M, M2 can actually happen, take a = -1, define
(2x² – 1
(-1 < x < 0),
f(x) = {x? – 1
(0<x< 0),
(x² + 1
and show that Mo =1, M, =4, M2=4.
Does Mi <4M, M2 hold for vector-valued functions too?
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