6. Suppose f and g are functions with domain R. If both f and g are even but f+g is odd, then prove that g(x) = -f(x) for all r ER.
6. Suppose f and g are functions with domain R. If both f and g are even but f+g is odd, then prove that g(x) = -f(x) for all r ER.
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
![6. Suppose f and g are functions with domain R. If both f and g are even but f+g is odd,
then prove that g(x) = -f(x) for all r ER.
%3D
7. Suppose neither lim f(x) nor lim g(x) exists. Show that lim f(x)g(x)] may exist.
x – 4
8. Prove that lim Va = 2 using the e-d-definition.
Hint: Va - 2 =
%3D
VI +2
9. Suppose g is a function continuous at c and g(c) > 0. Prove that there exists 8>0 such
that g(x) >0 for all r E (c- 8, c+ 8).
10. Consider the function
if r E Q
f (x) =
if x Q.
Prove that f'(0) = 0.
Hint: The fact stated in Question 10 of Problem Set 2 is useful.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe782b0f-9843-456a-9620-1984b4d9f5ef%2F946e6828-9e96-4620-9fe6-3a6a7c297091%2Fzemed8v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. Suppose f and g are functions with domain R. If both f and g are even but f+g is odd,
then prove that g(x) = -f(x) for all r ER.
%3D
7. Suppose neither lim f(x) nor lim g(x) exists. Show that lim f(x)g(x)] may exist.
x – 4
8. Prove that lim Va = 2 using the e-d-definition.
Hint: Va - 2 =
%3D
VI +2
9. Suppose g is a function continuous at c and g(c) > 0. Prove that there exists 8>0 such
that g(x) >0 for all r E (c- 8, c+ 8).
10. Consider the function
if r E Q
f (x) =
if x Q.
Prove that f'(0) = 0.
Hint: The fact stated in Question 10 of Problem Set 2 is useful.
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