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 a e R.Suppose neither lim f(x) nor lim g(æ) exists. Show that lim[f(x)g(x)] may exist.x - 4Prove that lim Va = 2 using the e-8-definition.Hint: Va – 2Va +2Suppose g is a function continuous at c and g(c) > 0. Prove that there exists & > 0 suchthat g(x) > 0 for all æ € (c – 6, c + 8).Consider the functionif æ € Qf(x) =-22 if a ¢ Q.Prove that f'(0) = 0.Hint: The fact stated in Question 10 of Problem Set 2 is useful.Consider the function f(x) = Vr on some closed interval and by applying the MeanValue Theorem, show that3253 < V82 <108(Hardest) Suppose f is differentiable everywhere and f'(x) > 0 for all numbers a exceptfor a single number d. Prove that f is always strictly increasing.

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Answered: Suppose f and g are functions with…

Suppose f and g are functions with domain R. If both f and g are even but ƒ +g is odd, then prove that g(x) = – f(x) for all x € R.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

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 a e R.
Suppose neither lim f(x) nor lim g(æ) exists. Show that lim[f(x)g(x)] may exist.
x - 4
Prove that lim Va = 2 using the e-8-definition.
Hint: Va – 2
Va +2
Suppose g is a function continuous at c and g(c) > 0. Prove that there exists & > 0 such
that g(x) > 0 for all æ € (c – 6, c + 8).
Consider the function
if æ € Q
f(x) =
-22 if a ¢ Q.
Prove that f'(0) = 0.
Hint: The fact stated in Question 10 of Problem Set 2 is useful.
Consider the function f(x) = Vr on some closed interval and by applying the Mean
Value Theorem, show that
325
3 < V82 <
108
(Hardest) Suppose f is differentiable everywhere and f'(x) > 0 for all numbers a except
for a single number d. Prove that f is always strictly increasing.

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