【数学I】例題3.2.1：2次関数の最大・最小（One More）★

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% 例題I3.2.1：2次関数の最大・最小（One More）★
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次の2次関数に最大値，最小値があればそれを求めよ．

(1) $y=2x^2-4x+5$

(2) $y=-5x^2+10x$

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(1)

\begin{align*}
y&=2x^2-4x+5\\
&=2(x^2-2x)+5\\
&=2\{(x-1)^2-1^2\}+5\\
&=2(x-1)^2+3
\end{align*}

よって，$x=1$のとき，最小値$3$，最大値はない．

(2)

\begin{align*}
y&=-5x^2+10x\\
&=-5(x^2-2x)\\
&=-5\{(x-1)^2-1^2\}\\
&=-5(x-1)^2+5
\end{align*}

よって，$x=1$のとき，最大値$5$，最小値はない．

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% 問題I3.2.1：2次関数の最大・最小（One More）★
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次の2次関数に最大値，最小値があればそれを求めよ．

(1) $y=-3x^2+6x+2$

(2) $y=4x^2+8x-1$

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% 問題I3.2.1の解答（One More）★
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(1)

\begin{align*}
y&=-3x^2+6x+2\\
&=-3(x^2-2x)+2\\
&=-3\{(x-1)^2-1^2\}+2\\
&=-3(x-1)^2+5
\end{align*}

よって，$x=1$のとき，最大値$5$，最小値はない．

(2)

\begin{align*}
y&=4x^2+8x-1\\
&=4(x^2+2x)-1\\
&=4\{(x+1)^2-1^2\}-1\\
&=4(x+1)^2-5
\end{align*}

よって，$x=-1$のとき，最小値$-5$，最大値はない．

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% 例題I3.2.1の側注：2次関数の最大・最小（One More）★
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% 側注1
平方完成する．

% 側注2
グラフは下に凸の放物線で，頂点は$(1,3)$である．$y$の値はいくらでも大きくなるから，最大値はない．

% 側注3
平方完成する．

% 側注4
グラフは上に凸の放物線で，頂点は$(1,5)$である．$y$の値はいくらでも小さくなるから，最小値はない．

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% 問題I3.2.1の側注：2次関数の最大・最小（One More）★
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% 側注1
平方完成する．

% 側注2
平方完成する．

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% 例題I3.2.1の考え方・補足など：2次関数の最大・最小（One More）★
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% 考え方
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$y=ax^2+bx+c$の形の式を変形（平方完成）して，標準形$y=a(x-p)^2+q$の形にする．

$y=ax^2+bx+c=a(x-p)^2+q$は，

$a>0$のとき，$x=p$で最小値$q$，最大値はない．

$a<0$のとき，$x=p$で最大値$q$，最小値はない．

$a>0$のとき

$a<0$のとき

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% 注意
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最大値，最小値をとるときの$x$の値は「必ず記すべきである」という考え方もある．