Suppose you are given the following theorem: Let f, g be two C² functions on R'. Fix x € R' and h > 0, and consider the closed interval [æ, x + h]. Then for every y € [x, x + h] there exists { E (x, x + h) such that {f (y) – [f (x) + f' (x)(y – x)]}g"(£) = f"(£){g(y) – [g(x) + g'(x)(y – x)]}. Use this theorem to prove the 2nd order Mean Value Theorem. That is prove that if f is C² on R' then f (x + h) = f(x) +h f' (x) + f" (x + Oh) for some 0 € (0, 1). Hint: let g(y) = (y – x)². %3D

Suppose you are given the following theorem: Let f, g be two C² functions on R'. Fix x € R' and h > 0, and consider the closed interval [æ, x + h]. Then for every y € [x, x + h] there exists { E (x, x + h) such that {f (y) – [f (x) + f' (x)(y – x)]}g"(£) = f"(£){g(y) – [g(x) + g'(x)(y – x)]}. Use this theorem to prove the 2nd order Mean Value Theorem. That is prove that if f is C² on R' then f (x + h) = f(x) +h f' (x) + f" (x + Oh) for some 0 € (0, 1). Hint: let g(y) = (y – x)². %3D

Name: Suppose you are given the following theorem: Let f, g be two C² funct
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Description: Suppose you are given the following theorem: Let f, g be two C² functionson R'. Fix x € R' and h > 0, and consider the closed interval [æ, x + h].Then for every y € [x, x + h] there exists { E (x, x + h) such that{f (y) – [f (x) + f' (x)(y – x)]}g"(

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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Suppose you are given the following theorem: Let f, g be two C² functions
on R'. Fix x € R' and h > 0, and consider the closed interval [æ, x + h].
Then for every y € [x, x + h] there exists { E (x, x + h) such that
{f (y) – [f (x) + f' (x)(y – x)]}g"(£) = f"(£){g(y) – [g(x) + g'(x)(y – x)]}.
Use this theorem to prove the 2nd order Mean Value Theorem. That is
prove that if f is C² on R' then f (x + h) = f(x) +h f' (x) + f" (x + Oh)
for some 0 € (0, 1). Hint: let g(y) = (y – x)².
%3D

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Suppose you are given the following theorem: Let f, g be two C² functions on R'. Fix x € R' and h > 0, and consider the closed interval [æ, æ + h]. Then for every y E [x, x + h] there exists & E (x, x +h) such that {f(y) – [f (x) + f' (x)(y – x)]}g"(£) = f"(E){g(y) – [g(x)+g'(x)(y – x)]}. Use this theorem to prove the 2"d order Mean Value Theorem. That is prove that if f is C² on R' then f (x + h) = f(x) +hf' (x) + f"(x + 0h) for some 0 E (0, 1). Hint: let g(y) = (y – x)².