二次函數的Max Min 200811251023

跳轉到: 導航, 搜索

屬性

$\begin{align} & x\in \mathbb{R} \\ & f\left( x \right)=\left( x^{2}+4x+5 \right)\left( x^{2}+4x+2 \right)+2x^{2}+8x+1, \\ \end{align}$

當
$\begin{align} & x=m, \\ & y=Minf\left( x \right)=M, \\ & \left( m,M \right) \\ \end{align}$

$f\left( x \right)=\left( x^{2}+4x+5 \right)\left( x^{2}+4x+2 \right)+2x^{2}+8x+1,$
設
$\begin{align} & a=x^{2}+4x=\left( x^{2}+4x+4 \right)-4 \\ & =\left( x+2 \right)^{2}-4 \\ \end{align}$
當 $x=-2\,$ 時， $a\,$ 有最小值
$\Rightarrow \left\{ \begin{align} & x=-2=m \\ & \min a=-4 \\ \end{align} \right.$
將 $a=x^{2}+4x\,$ 代入 $f\left( x \right)$
$\begin{align} & y=\left( a+5 \right)\left( a+2 \right)+2a+1 \\ & =a^{2}+7a+10+2a+1 \\ & =a^{2}+9a+11 \\ & =\left( a+\frac{9}{2} \right)^{2}-\frac{81}{4}+11 \\ & =\left( a+\frac{9}{2} \right)^{2}-\frac{37}{4} \\ \end{align}$
當 $a=-4\,$ 時， $f\left( x \right)$ 有最小值
$\Rightarrow y=\left( \frac{1}{2} \right)^{2}-\frac{37}{4}=\frac{-36}{4}=-9$
$\left( m,M \right)=\left( -2,-9 \right)\#$