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4. Prove that cos x cos y ≤ x - y for any x, y.

4. Prove that cos x cos y ≤ x - y for any x, y.

Advanced Engineering Mathematics
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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### Problems in Calculus #### Problem 4: Prove that \(|\cos x - \cos y| \leq |x - y|\) for any \(x, y\). **Hint:** Use the Mean Value Theorem (MVT). #### Problem 5: Assume that \(f\) is differentiable on \(\mathbb{R}\) such that \(f(0) = 0\), \(f(1) = 1\), and \(f(2) = 1\). Show that there exists some \(x\) in \((0, 2)\) such that \(f'(x) = \frac{1}{10}\). **Hint:** Utilize the Mean Value Theorem (MVT) and the Intermediate Value Theorem (IVT) for derivatives.
Transcribed Image Text:### Problems in Calculus #### Problem 4: Prove that \(|\cos x - \cos y| \leq |x - y|\) for any \(x, y\). **Hint:** Use the Mean Value Theorem (MVT). #### Problem 5: Assume that \(f\) is differentiable on \(\mathbb{R}\) such that \(f(0) = 0\), \(f(1) = 1\), and \(f(2) = 1\). Show that there exists some \(x\) in \((0, 2)\) such that \(f'(x) = \frac{1}{10}\). **Hint:** Utilize the Mean Value Theorem (MVT) and the Intermediate Value Theorem (IVT) for derivatives.
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