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 letF(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 maximumvalue 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 minimumvalue at c if and only if F has a local minimum value at c.

Given:A function on the interval such that:is defined for all . for all .The function is defined…

Answered: F(x) f(x) 1 + f²(x)*

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F(x) f(x) 1 + f²(x)*

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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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.

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