简介

若已知 $f(x)>0$，则构造函数 $F(x)$，使 $F^{\prime }(x)=\phantom{0}$$\phantom{0}\text{某正数}\times f(x)$，利用 $F(x)\uparrow$ 解题。$f(x)<0$ 同理。

例 1

已知 $f(x)+\phantom{0}$$\phantom{0}{f}^{\prime }(x)>0$，$f(0)=\phantom{0}$$\phantom{0}1$，求证当 $x\in [0,\phantom{0}$$\phantom{0}+\phantom{0}$$\phantom{0}\mathrm{\infty })$ 时 $f(x)\ge {\displaystyle \frac{1}{e^x}}$。

证明

构造 $g(x)=\phantom{0}$$\phantom{0}{e}^{x}f(x)$，则

$$g^{\prime }(x)=e^{x}f(x)+e^x{f}^{\prime }(x)=e^x[f(x)+f^{\prime }(x)]>0$$

$\therefore x\in [0,\phantom{0}$$\phantom{0}+\phantom{0}$$\phantom{0}\mathrm{\infty })$ 时，$g(x)\uparrow$，$g(x)\ge g(0)$，即 $e^{x}f(x)\ge e^{0}f(0)=\phantom{0}$$\phantom{0}1$。

$\therefore f(x)\ge {\displaystyle \frac{1}{e^x}}$。

基本构造

和差构造

• $f^{\prime }+\phantom{0}$$\phantom{0}{g}^{\prime }>0\Rightarrow F=\phantom{0}$$\phantom{0}f+\phantom{0}$$\phantom{0}g$
• $f^{\prime }-\phantom{0}$$\phantom{0}{g}^{\prime }>0\Rightarrow F=\phantom{0}$$\phantom{0}f-\phantom{0}$$\phantom{0}g$

积商构造

• $f^{\prime }g+\phantom{0}$$\phantom{0}f{g}^{\prime }>0\Rightarrow F=\phantom{0}$$\phantom{0}fg$
• $f^{\prime }g-\phantom{0}$$\phantom{0}f{g}^{\prime }>0\Rightarrow F=\phantom{0}$$\phantom{0}{\displaystyle \frac{f}{g}}(g\ne \phantom{0}$$\phantom{0}0)$

变形

含 $x$ 形构造

• $x{f}^{\prime }(x)+\phantom{0}$$\phantom{0}nf(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{x}^{n}f(x)$
• $x{f}^{\prime }(x)-\phantom{0}$$\phantom{0}nf(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{\displaystyle \frac{f(x)}{x^n}}(x\ne \phantom{0}$$\phantom{0}0)$

含 $e$ 形构造

• $f^{\prime }(x)+\phantom{0}$$\phantom{0}nf(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{e}^{nx}f(x)$
• $f^{\prime }(x)-\phantom{0}$$\phantom{0}nf(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{\displaystyle \frac{f(x)}{e^{nx}}}$

三角构造

• $f(x)+\phantom{0}$$\phantom{0}{f}^{\prime }(x)\tan x>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}\sin xf(x)$

• $f(x)-\phantom{0}$$\phantom{0}{f}^{\prime }(x)\tan x>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{\displaystyle \frac{f(x)}{\sin x}}(\sin x\ne \phantom{0}$$\phantom{0}0)$

• $f(x)\tan x+\phantom{0}$$\phantom{0}{f}^{\prime }(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{\displaystyle \frac{f(x)}{\cos x}}(\cos x\ne \phantom{0}$$\phantom{0}0)$

• $f(x)\tan x-\phantom{0}$$\phantom{0}{f}^{\prime }(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}\cos xf(x)$

对数形构造

• $f^{\prime }(x)+\phantom{0}$$\phantom{0}\ln af(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{a}^{x}f(x)$
• $f^{\prime }(x)-\phantom{0}$$\phantom{0}\ln af(x)>0\Rightarrow F(x)=\phantom{0}$$\phantom{0}{\displaystyle \frac{f(x)}{a^x}}$