构造函数典例及其保号性思路

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f(x) is continuous on[0,1]and f(x)0xf(t)dtProve thata,b,if:0<a<b<1,then:abf(x)dx0\begin{aligned} &\text f(x)\ is\ continuous\ on[0,1],and\ f(x)\leq \displaystyle \int^{x}_{0}{f(t)dt}\\ &\text Prove\ that:\forall a,b,if:0<a<b<1,then:\int^{b}_{a}{f(x)dx} \leq 0 \end{aligned}

构造法

Assume thatg(x)=ex0xf(t)dtTheng(x)=[f(x)0xf(t)dt]ex\begin{aligned} &Assume\ that\quad g(x)=e^{-x} \displaystyle \int ^x_0{f(t)}dt\\ &Then\quad g^\prime(x)=[f(x)- \displaystyle \int ^x_0f(t)dt]e^{-x} \end{aligned}

Which is known:f(x)0xf(t)dtf(x)\leq \displaystyle \int^{x}_{0}{f(t)dt},so g(x)0g'(x) \leq 0

g(x)0,g(0)=0g(x)is on[0,1]so we have g(x)0 ,being true.And for exon[0,1]ex>0On[0,1]we have0xf(t)dt0a,b,0<a<b<1,abf(x)dx0\begin{aligned} &\because g'(x) \leq 0,g(0)=0\\ \\ &\therefore g(x)\text is\ on[0,1],\\ \\ &\text{so we have g(x)} \leq0\ \text, being \ true. \\ \\ And\ for&\ e^{-x} on[0,1],e^{-x}>0\\ \\ &\therefore \text On[0,1],we\ have\displaystyle \int ^x_0{f(t)}dt\leq0\\ \\ &\therefore \forall a,b,0<a<b<1,\\ &\quad\int^{b}_{a}{f(x)dx} \leq 0 \end{aligned}


保号性思路

构造法只能针对特定类型的题目,一般情况下会采用从定义的方法入手

参考视频链接

文章目录
  1. 构造法
  2. 保号性思路
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