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Let $f(\mathrm{x})=\mathrm{x}+\frac{1}{\mathrm{x}}$, when $\mathrm{x} \in(0,1)$. Then which one of the following is correct?
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Verified Answer
The correct answer is:
$f(\mathrm{x})$ decreases in the interval
$f(x)=x+\frac{1}{x}$
$f^{\prime}(x)=1-\frac{1}{x^{2}}=\frac{x^{2}-1}{x^{2}}=\frac{(x-1)(x+1)}{x^{2}}$
For $x \in(0,1), f^{\prime}(x) < 0$
$\Rightarrow \mathrm{f}(\mathrm{x})$ decreases
$f^{\prime}(x)=1-\frac{1}{x^{2}}=\frac{x^{2}-1}{x^{2}}=\frac{(x-1)(x+1)}{x^{2}}$
For $x \in(0,1), f^{\prime}(x) < 0$
$\Rightarrow \mathrm{f}(\mathrm{x})$ decreases
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