3. Suppose that f: [a, b] → R is differentiable, and that for all a [a, b] we have f'(x) #0. Prove that f is one-to-one on [a, b]. Provide an example to show that the converse need not be true.
3. Suppose that f: [a, b] → R is differentiable, and that for all a [a, b] we have f'(x) #0. Prove that f is one-to-one on [a, b]. Provide an example to show that the converse need not be true.
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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![**Problem Statement:**
3. Suppose that \( f : [a, b] \rightarrow \mathbb{R} \) is differentiable, and that for all \( x \in [a, b] \) we have \( f'(x) \neq 0 \). Prove that \( f \) is one-to-one on \([a, b]\).
**Follow-Up Exploration:**
Provide an example to show that the converse need not be true.
**Explanation:**
This problem explores the relationship between differentiability and injectivity. In particular, it asks you to prove that if a function is differentiable on the interval \([a, b]\) and its derivative is nonzero throughout that interval, the function must be one-to-one (injective) on that interval. Additionally, it invites you to consider the converse scenario—whether injectivity necessarily implies that the derivative is always nonzero—and to provide a counterexample if it does not.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2F6d924cbe-ebcb-474f-9130-43ae6543b302%2Fqpcnn0k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
3. Suppose that \( f : [a, b] \rightarrow \mathbb{R} \) is differentiable, and that for all \( x \in [a, b] \) we have \( f'(x) \neq 0 \). Prove that \( f \) is one-to-one on \([a, b]\).
**Follow-Up Exploration:**
Provide an example to show that the converse need not be true.
**Explanation:**
This problem explores the relationship between differentiability and injectivity. In particular, it asks you to prove that if a function is differentiable on the interval \([a, b]\) and its derivative is nonzero throughout that interval, the function must be one-to-one (injective) on that interval. Additionally, it invites you to consider the converse scenario—whether injectivity necessarily implies that the derivative is always nonzero—and to provide a counterexample if it does not.
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