(6) Prove Darboux's Theorem: If f is differentiable on all of R and a and b are any real numbers such that a < b, and f'(a) < f'(b), then for every real number y such that f'(a) < y < f'(b), there is ace (a,b) such that f'(c) = y. (In other words, Darboux's Theorem says that the deritvative of a function satisfies the conclusion of the IVT.) Tips: First consider the case where f'(a) < 0, f'(b) > 0 and y = 0. For this case, by the Extreme Value Theorem f attains its minimum value on [a, b] at some point c e [a, b]. Prove that c + a and c + b (using that f'(a) < 0 and f'(b) > 0), and then apply the Min-Max Theorem. For the general case, apply the first case to the function h(x) := f(x) – yæ.
(6) Prove Darboux's Theorem: If f is differentiable on all of R and a and b are any real numbers such that a < b, and f'(a) < f'(b), then for every real number y such that f'(a) < y < f'(b), there is ace (a,b) such that f'(c) = y. (In other words, Darboux's Theorem says that the deritvative of a function satisfies the conclusion of the IVT.) Tips: First consider the case where f'(a) < 0, f'(b) > 0 and y = 0. For this case, by the Extreme Value Theorem f attains its minimum value on [a, b] at some point c e [a, b]. Prove that c + a and c + b (using that f'(a) < 0 and f'(b) > 0), and then apply the Min-Max Theorem. For the general case, apply the first case to the function h(x) := f(x) – yæ.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Proving Darboux's Theorem**
**Problem Statement:**
Prove Darboux's Theorem: If \( f \) is differentiable on all of \(\mathbb{R}\) and \( a \) and \( b \) are any real numbers such that \( a < b \), and \( f'(a) < f'(b) \), then for every real number \( y \) such that \( f'(a) < y < f'(b) \), there is a \( c \in (a, b) \) such that \( f'(c) = y \). In other words, Darboux's Theorem says that the derivative of a function satisfies the conclusion of the Intermediate Value Theorem (IVT).
**Tips for Proving the Theorem:**
1. **Initial Consideration:**
- First, consider the case where \( f'(a) < 0 \), \( f'(b) > 0 \), and \( y = 0 \).
- By the Extreme Value Theorem, \( f \) attains its minimum value on \([a, b]\) at some point \( c \in [a, b] \).
- Prove that \( c \neq a \) and \( c \neq b \) using the fact that \( f'(a) < 0 \) and \( f'(b) > 0 \).
- Apply the Min-Max Theorem.
2. **General Case:**
- For the general case, apply the first case to the function \( h(x) := f(x) - yx \).
This problem provides a conceptual understanding of how derivatives behave according to Darboux's Theorem, illustrating the Intermediate Value Property with derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba5a3306-2973-4825-a0c7-f3a4b662c857%2F7b4f4cbd-e15d-4e76-a5c2-46b0c08cf07f%2F98q8mni_processed.png&w=3840&q=75)
Transcribed Image Text:**Proving Darboux's Theorem**
**Problem Statement:**
Prove Darboux's Theorem: If \( f \) is differentiable on all of \(\mathbb{R}\) and \( a \) and \( b \) are any real numbers such that \( a < b \), and \( f'(a) < f'(b) \), then for every real number \( y \) such that \( f'(a) < y < f'(b) \), there is a \( c \in (a, b) \) such that \( f'(c) = y \). In other words, Darboux's Theorem says that the derivative of a function satisfies the conclusion of the Intermediate Value Theorem (IVT).
**Tips for Proving the Theorem:**
1. **Initial Consideration:**
- First, consider the case where \( f'(a) < 0 \), \( f'(b) > 0 \), and \( y = 0 \).
- By the Extreme Value Theorem, \( f \) attains its minimum value on \([a, b]\) at some point \( c \in [a, b] \).
- Prove that \( c \neq a \) and \( c \neq b \) using the fact that \( f'(a) < 0 \) and \( f'(b) > 0 \).
- Apply the Min-Max Theorem.
2. **General Case:**
- For the general case, apply the first case to the function \( h(x) := f(x) - yx \).
This problem provides a conceptual understanding of how derivatives behave according to Darboux's Theorem, illustrating the Intermediate Value Property with derivatives.
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