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how to prove that $f(x) = 2x\sin(\frac{1}{x}) - \cos (\frac{1}{x}) + 2$ is positive when $x\in (0,1]$.

I can see that by plotting.

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2 Answers 2

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Look first at $x\ge \frac{1}{2}$. Then $\sin(1/x)$ is positive, so our function is positive.

Look next at $0\lt x\lt \frac{1}{2}$. Then $2x\lt 1$. Thus $2x\sin(1/x)\gt -1$, so again our function is positive.

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  • $\begingroup$ nicholas: how do you justify the first line ? $\endgroup$
    – aaaaaa
    Apr 8, 2014 at 6:21
  • $\begingroup$ If $1/2\le x\le 1$, then $1\le \frac{1}{x}\le 2$. But $\sin t$ is positive in the whole interval $0\lt t\lt \pi$, and therefore in particular in the interval $[1,2]$. $\endgroup$ Apr 8, 2014 at 6:25
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Try looking at when each of the terms is positive/negative over the given interval. Then show that the positives have a larger absolute value than the negatives for any value in $[0,1]$.

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