Use Question 1 and find the critical points for the function 2²³ 5 -4+1 3 1 + (+)² for x = [1,1]. Which of these critical points are local maxima? F(x)= 2

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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Just answer the 2nd question
Let f be a function on the interval [a, b] such that:
f'(x) is defined for all x € [a, b]
|f(x) < 1 for all x = [a, b]
●
and let
F(x) =
f(x)
1 + f2²(x)
Prove the following statements:
a) The functions f and F have the same critical points on the interval [a, b].
b) Suppose that c € [a, b] is a common critical point of f and F. Prove that f has a local maximum
value at c if and only if F has a local maximum value at c.
c) It is also true that if c € [a, b] is a common critical point of f and F, then f has a local minimum
value at c if and only if F has a local minimum value at c.
Transcribed Image Text:Let f be a function on the interval [a, b] such that: f'(x) is defined for all x € [a, b] |f(x) < 1 for all x = [a, b] ● and let F(x) = f(x) 1 + f2²(x) Prove the following statements: a) The functions f and F have the same critical points on the interval [a, b]. b) Suppose that c € [a, b] is a common critical point of f and F. Prove that f has a local maximum value at c if and only if F has a local maximum value at c. c) It is also true that if c € [a, b] is a common critical point of f and F, then f has a local minimum value at c if and only if F has a local minimum value at c.
2. Use Question 1 and find the critical points for the function
x3 -4+1
5
F(x)
=
2
1 + ( ²³² − ² + 3 )
for x = [1,1]. Which of these critical points are local maxima?
Transcribed Image Text:2. Use Question 1 and find the critical points for the function x3 -4+1 5 F(x) = 2 1 + ( ²³² − ² + 3 ) for x = [1,1]. Which of these critical points are local maxima?
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