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[函数] 来自人教论坛的……太简单懒得想标题

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kuing posted 2013-9-16 13:16 |Read mode
链接:bbs.pep.com.cn/forum.php?mod=viewthread&tid=2878322
发贴ID:ytxiehua2008
求证:$\forall x\in\Bbb R^+$, $\displaystyle
\frac{1-x^2-(x^2+x)\ln x}{e^x}<\frac43$。
原不等式等价于
\[(x+1)(1-x-x\ln x)<\frac43e^x,\]
由 $e^x\geqslant x+1$ 可知只要证
\[x+x\ln x>-\frac13,\]
令 $f(x)=x+x\ln x$,则
\[f'(x)=2+\ln x \riff f(x)_{\min}=f(e^{-2})=-e^{-2}>-\frac13,\]
得证。

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