# 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

${\displaystyle I:=[a,b]}$ 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。

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

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