Prove that $$ f(x)=x+(1/2)\cos(x) $$ is strictly increasing.
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1Have you tried deriving it? – 5xum Feb 25 '14 at 07:36
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I dont know-how to derivate – Denis Feb 25 '14 at 07:39
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Then the job may be tougher than what I originaly thought. Is this some sort of homework? Because the solution is trivial once you know a few things about derivatives. – 5xum Feb 25 '14 at 07:41
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Do you know how to show that $|\cos x - \cos y| \le |x-y|$? – copper.hat Feb 25 '14 at 07:47
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Nope i dont know – Denis Feb 25 '14 at 08:18
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How about $|\sin x| \le |x|$? – copper.hat Feb 25 '14 at 08:19
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1What is your definition of cos? – ljfa Feb 25 '14 at 08:29
1 Answers
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$f'(x) = 1-{1 \over 2} \sin x \ge {1 \over 2} $ for all $x$.
copper.hat
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The derivative of $\cos$ is $-\sin$ however, so it is $1 -\frac12 \sin x$ – ljfa Feb 25 '14 at 08:00
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Please don't mind...but the OP said that he doesn't know derivatives yet. – Hawk Feb 25 '14 at 08:34