# 保西奴中間點定理

## 根定位定理

• 如果${\displaystyle f(m_{1})>0}$，設${\displaystyle a_{2}=a,b_{2}=m_{1}}$${\displaystyle m_{2}={\frac {1}{2}}(a_{2}+b_{2})}$就係${\displaystyle I_{2}:=[a_{2},b_{2}]}$間距嘅中間點。
• 如果${\displaystyle f(m_{1})<0}$，設${\displaystyle a_{2}=m_{1},b_{2}=b_{1}}$${\displaystyle m_{2}={\frac {1}{2}}(a_{2}+b_{2})}$就係${\displaystyle I_{2}:=[a_{2},b_{2}]}$間距嘅中間點。

• 如果${\displaystyle f(m_{k})=0}$，咁${\displaystyle c=m_{k}}$，可以收工。
• 如果${\displaystyle f(m_{k})>0}$，設${\displaystyle a_{k+1}=a_{k},b_{k+1}=m_{k}}$${\displaystyle m_{k+1}={\frac {1}{2}}(a_{k+1}+b_{k+1})}$就係${\displaystyle I_{k+1}:=[a_{k+1},b_{k+1}]}$間距嘅中間點。
• 如果${\displaystyle f(m_{k})<0}$，設${\displaystyle a_{k+1}=m_{k},b_{k+1}=b_{k}}$${\displaystyle m_{k+1}={\frac {1}{2}}(a_{k+1}+b_{k+1})}$就係${\displaystyle I_{k+1}:=[a_{k+1},b_{k+1}]}$間距嘅中間點。

${\displaystyle I_{n}:=[a_{n},b_{n}],\forall n\in \mathbb {N} }$，同時間會有${\displaystyle f(a_{n})<0}$同埋${\displaystyle f(b_{n})>0}$

## 中間點定理

${\displaystyle g(x):=f(x)-k}$

${\displaystyle g(a)=f(a)-k}$${\displaystyle g(b)=f(b)-k}$

${\displaystyle f(c)=k}$

${\displaystyle g(x):=-f(x)+k}$

${\displaystyle g(a)=-f(a)+k}$${\displaystyle g(b)=-f(b)+k}$

${\displaystyle f(c)=k}$

${\displaystyle I:=[a,b]}$係一個關閉又被綁定嘅間距，${\displaystyle f:I\to \mathbb {R} }$係喺${\displaystyle I}$上面連續嘅。

## 應用

${\displaystyle I}$係一個關閉又被綁定嘅間距，${\displaystyle f:I\to \mathbb {R} }$係喺${\displaystyle I}$上面連續嘅。咁${\displaystyle S:=\{f(x):x\in I\}}$都係一個關閉又被綁定嘅間距。

${\displaystyle s_{\star }:=\inf S}$${\displaystyle s^{\star }:=\sup S}$

${\displaystyle s\in [s_{\star },s^{\star }]}$