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3. We can use Rolle's Theorem to prove a generalized Mean Value Theorem. Theorem 1. Let a

3. We can use Rolle's Theorem to prove a generalized Mean Value Theorem. Theorem 1. Let a

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3. We can use Rolle's Theorem to prove a generalized Mean Value Theorem. Theorem 1. Let a <b. Let f and g be two functions which are continuous on [a, b] and differentiable on (a, b). Assume g(b) g(a) and g'(x) 0 for any rЄ (a, b). Then there exists & Є (a, b) such that f(b) f(a) g(b) - g(a) = f'(§) g'(§) (a) We can find a specific g(x) in the theorem to get the standard Mean Value Theorem. What is g(x) ? No need to justify your answer to this part. (b) Prove the theorem 1. Hint: Define a new function F(x) = f(x) g(a)]. - 16g(x) or H(x) = g(x)[f(b) − f(a)] − f(x)|g(b)— -
Transcribed Image Text:3. We can use Rolle's Theorem to prove a generalized Mean Value Theorem. Theorem 1. Let a <b. Let f and g be two functions which are continuous on [a, b] and differentiable on (a, b). Assume g(b) g(a) and g'(x) 0 for any rЄ (a, b). Then there exists & Є (a, b) such that f(b) f(a) g(b) - g(a) = f'(§) g'(§) (a) We can find a specific g(x) in the theorem to get the standard Mean Value Theorem. What is g(x) ? No need to justify your answer to this part. (b) Prove the theorem 1. Hint: Define a new function F(x) = f(x) g(a)]. - 16g(x) or H(x) = g(x)[f(b) − f(a)] − f(x)|g(b)— -
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