7.The goal of this problem is a proof of the Lagrange formof the remainder in Taylor's theorem (Corollary 8.13(b) in the book)that is analogous to the proof of the mean value theorem. Let I bean open interval, and let ƒ be a function for which f, f', f", ..., f(n+1)exist on I.(a) Prove the following:Lemma (Generalized Rolle's theorem). If a,b ≤ I are distinctand f(a) = f'(a) = f'(a)= ... = f(n)(a) = f(b) = 0, then there existsa point c = (a, b) or (b, a) such that f(n+1) (c) = 0. Hint. Induction.

Answered: Image /qna-images/answer/011a3bfe-8fc0-4d5a-91a2-b8c8e5e4f172.jpg

7. The goal of this problem is a proof of the Lagrange form of the remainder in Taylor's theorem (Corollary 8.13(b) in the book) that is analogous to the proof of the mean value theorem. Let I be an open interval, and let f be a function for which f, f', f", ..., f(n+1) exist on I. (a) Prove the following: Lemma (Generalized Rolle's theorem). If a, b = I are distinct and f(a) = f'(a) = f'(a) f(n)(a) = f(b) = 0, then there exists a point c = (a, b) or (b, a) such that f(n+¹)(c) = 0. Hint. Induction. = ... =

7. The goal of this problem is a proof of the Lagrange form of the remainder in Taylor's theorem (Corollary 8.13(b) in the book) that is analogous to the proof of the mean value theorem. Let I be an open interval, and let f be a function for which f, f', f", ..., f(n+1) exist on I. (a) Prove the following: Lemma (Generalized Rolle's theorem). If a, b = I are distinct and f(a) = f'(a) = f'(a) f(n)(a) = f(b) = 0, then there exists a point c = (a, b) or (b, a) such that f(n+¹)(c) = 0. Hint. Induction. = ... =

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.5: Applications

Problem 28EQ

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