📝 题目
例 1 求证: $f\left( x\right) = {\left( \arctan \frac{x}{1 - x}\right) }^{-\frac{1}{2}}$ 在(0,1)内单调下降.
💡 答案与解析
证 设 $\varphi \left( x\right) = \frac{x}{1 - x}$ ,则
$$ \varphi \left( x\right) > 0\;\left( {x \in \left( {0,1}\right) }\right) , $$
$$ {\varphi }^{\prime }\left( x\right) = {\left\{ \frac{1}{1 - x} - 1\right\} }^{\prime } = \frac{1}{{\left( 1 - x\right) }^{2}} > 0\;\left( {x \in \left( {0,1}\right) }\right) , $$
因此
$$ {\left\lbrack \arctan \varphi \left( x\right) \right\rbrack }^{\prime } = \frac{{\varphi }^{\prime }\left( x\right) }{1 + \varphi {\left( x\right) }^{2}} > 0\;\left( {x \in \left( {0,1}\right) }\right) . \tag{3.1} $$
又因为 $f\left( x\right) = {\left\lbrack \arctan \varphi \left( x\right) \right\rbrack }^{-\frac{1}{2}}$ ,所以
$$ {f}^{\prime }\left( x\right) = - \frac{1}{2}{\left\lbrack \arctan \varphi \left( x\right) \right\rbrack }^{-\frac{3}{2}}{\left\lbrack \arctan \varphi \left( x\right) \right\rbrack }^{\prime }. \tag{3.2} $$
联合 (3.1) 与 (3.2) 式知 ${f}^{\prime }\left( x\right) < 0\left( {x \in \left( {0,1}\right) }\right)$ ,从而 $f\left( x\right)$ 在(0,1)内单调下降.
评注 值得注意的是,在证题过程中引进中间函数 $\varphi \left( x\right)$ ,使表述显得简捷.