(a) Let f [a, b] → R be a function such that f is continuous on [a, b] and differentiable on Ja, b[. Assume, furthermore, that f(a) ≥ 0 and that f'(x) > 0 for all x Ela, b[. Use the mean value theorem to prove that f(x) > 0 for all x ]a, b]. No other method will be accepted! (b) Prove that sinx < x for all x > 0. You may use, without proof, the fact that sinx is differentiable on R with (sin x)' = cosx. Hint: Prove first that sin x < x for all x €]0,].
(a) Let f [a, b] → R be a function such that f is continuous on [a, b] and differentiable on Ja, b[. Assume, furthermore, that f(a) ≥ 0 and that f'(x) > 0 for all x Ela, b[. Use the mean value theorem to prove that f(x) > 0 for all x ]a, b]. No other method will be accepted! (b) Prove that sinx < x for all x > 0. You may use, without proof, the fact that sinx is differentiable on R with (sin x)' = cosx. Hint: Prove first that sin x < x for all x €]0,].
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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Question
![(a) Let f [a, b] → R be a function such that f is continuous on [a, b] and differentiable on
Ja, b[. Assume, furthermore, that f(a) ≥ 0 and that f'(x) > 0 for all x Ela, b[. Use the
mean value theorem to prove that f(x) > 0 for all x ɛ]a,b]. No other method will be
accepted!
(b) Prove that sinx < x for all x > 0. You may use, without proof, the fact that sinx is
differentiable on R with (sin x)' = cos x.
Hint: Prove first that sin x < x for all x € ]0,5].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F69a7ae5f-ee7a-4c98-a4a2-cdd0b4d21034%2Fc706984c-5339-4a00-98aa-e91d59ece9f1%2F68dk77h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Let f [a, b] → R be a function such that f is continuous on [a, b] and differentiable on
Ja, b[. Assume, furthermore, that f(a) ≥ 0 and that f'(x) > 0 for all x Ela, b[. Use the
mean value theorem to prove that f(x) > 0 for all x ɛ]a,b]. No other method will be
accepted!
(b) Prove that sinx < x for all x > 0. You may use, without proof, the fact that sinx is
differentiable on R with (sin x)' = cos x.
Hint: Prove first that sin x < x for all x € ]0,5].
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