Suppose you are given the following theorem: Let f, g be two C² functions on R'. Fix a € 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) – [S(x) + f' (x)(y – x)]}g"(£) = f"(£){g(y) – [g(x) + g'(x)(y – x)]}. Use this theorem to prove the 2"nd 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 + 0h) for some 0 E (0, 1). Hint: let g(y) = (y – x)².
Suppose you are given the following theorem: Let f, g be two C² functions on R'. Fix a € 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) – [S(x) + f' (x)(y – x)]}g"(£) = f"(£){g(y) – [g(x) + g'(x)(y – x)]}. Use this theorem to prove the 2"nd 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 + 0h) for some 0 E (0, 1). Hint: let g(y) = (y – x)².
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 a € 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) – [S(x) + f' (x)(y – x)]}g"(£) = f"(£){g(y) – [g(x) + g'(x)(y – x)]}.
Use this theorem to prove the 2"nd 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 + 0h)
for some 0 E (0, 1). Hint: let g(y) = (y – x)².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfe9f85b-de9f-42f6-a5b9-dccaa3428661%2F19599dbd-6569-4499-86cc-2c0d4fc8291c%2F8iayu5m.png&w=3840&q=75)
Transcribed Image Text:Suppose you are given the following theorem: Let f, g be two C² functions
on R'. Fix a € 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) – [S(x) + f' (x)(y – x)]}g"(£) = f"(£){g(y) – [g(x) + g'(x)(y – x)]}.
Use this theorem to prove the 2"nd 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 + 0h)
for some 0 E (0, 1). Hint: let g(y) = (y – x)².
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