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Let ƒ : [a, b] → R be differentiable on [a, b]. Let λ = R be such that ƒ'(a) < \ < f'(b). (i) Let h : [a, b] → R be defined as h(x) = f(x) — λx. Prove that minimum of h is not achieved at a or b. (ii) Use (i) to prove that there exists x € [a, b] such that f'(x) = \.

Let ƒ : [a, b] → R be differentiable on [a, b]. Let λ = R be such that ƒ'(a) < \ < f'(b). (i) Let h : [a, b] → R be defined as h(x) = f(x) — λx. Prove that minimum of h is not achieved at a or b. (ii) Use (i) to prove that there exists x € [a, b] such that f'(x) = \.

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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Could you explian how to prove this in detail?

Let f [a, b] → R be differentiable on [a, b]. Let A ER be such that f'(a) < \< ƒ'(b). (i) Let h : [a, b] → R be defined as h(x) = f(x) — λx. Prove that minimum of h is not achieved at a or b. = = \. (ii) Use (i) to prove that there exists x = [a, b] such that f'(x) =
Transcribed Image Text:Let f [a, b] → R be differentiable on [a, b]. Let A ER be such that f'(a) < \< ƒ'(b). (i) Let h : [a, b] → R be defined as h(x) = f(x) — λx. Prove that minimum of h is not achieved at a or b. = = \. (ii) Use (i) to prove that there exists x = [a, b] such that f'(x) =
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