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(b) Prove that if a, b E I are distinct, then there exists a c strictly between a and b such that Tn(t) = n f(b) = Σ k=0 in the following way. For each t € I, let f(k) (a) k! n ƒ(²) (a) (b − a)* + f(k) k! f(n+1) (c) (n + 1)! -(b − a) n+¹ Rn (b) (b-a)n +1 (t-a)n + ·(t-a)k, Rn(t) = f(t)-Tn(t), g(t) = Tn(t)+· k=0 Apply the generalized Rolle's theorem lemma to h = f - g.

(b) Prove that if a, b E I are distinct, then there exists a c strictly between a and b such that Tn(t) = n f(b) = Σ k=0 in the following way. For each t € I, let f(k) (a) k! n ƒ(²) (a) (b − a)* + f(k) k! f(n+1) (c) (n + 1)! -(b − a) n+¹ Rn (b) (b-a)n +1 (t-a)n + ·(t-a)k, Rn(t) = f(t)-Tn(t), g(t) = Tn(t)+· k=0 Apply the generalized Rolle's theorem lemma to h = f - g.

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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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.
Transcribed Image Text: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.
(b) Prove that if a, b = I are distinct, then there exists a c strictly between a and b such that f(b) = n f(k) (a) k! (b − a)k + f(n+1) (c) (n + 1)! (b − a)n +¹ k=0 in the following way. For each t € I, let n Tn(t) = f(a) (t-a)*, R₁(t) = f(t)-Tn(t), _g(t) = Tn(t)+ k! k=0 Apply the generalized Rolle's theorem lemma to h = f - g. Rn (b) (b − a)n + ¹ (t-a)n+1
Transcribed Image Text:(b) Prove that if a, b = I are distinct, then there exists a c strictly between a and b such that f(b) = n f(k) (a) k! (b − a)k + f(n+1) (c) (n + 1)! (b − a)n +¹ k=0 in the following way. For each t € I, let n Tn(t) = f(a) (t-a)*, R₁(t) = f(t)-Tn(t), _g(t) = Tn(t)+ k! k=0 Apply the generalized Rolle's theorem lemma to h = f - g. Rn (b) (b − a)n + ¹ (t-a)n+1
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