【数学I】例題3.2.4：定義域が拡大するときの最大・最小（One More）★★★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.2.4：定義域が拡大するときの最大・最小（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$a>0$とする．関数$f(x)=x^2-4x+6(0\leqq x\leqq a)$について，$f(x)$の最大値を求めよ．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.2.4の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\[
f(x)=x^2-4x+6=(x-2)^2+2
\]

$y=f(x)$のグラフは下に凸の放物線で，軸は直線$x=2$

(i) $0<a<4$のとき

グラフは右の図のようになる．

$x=0$のとき最大となり，最大値$f(0)=6$

(ii) $a=4$のとき

グラフは右の図のようになる．

$x=0,4$のとき最大となり，最大値$f(0)=f(4)=6$

(iii) $4<a$のとき

グラフは右の図のようになる．

$x=a$のとき最大となり，最大値$f(a)=a^2-4a+6$

よって，(i)〜(iii)より，

\[
\{\begin{array}{ll}0<a<4\text{のとき，}&x=0\text{で最大値}6\\a=4\text{のとき，}&x=0,4\text{で最大値}6\\4<a\text{のとき，}&x=a\text{で最大値}a^2-4a+6\end{array}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.2.4：定義域が拡大するときの最大・最小（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1) $a>0$とする．関数$f(x)=x^2-4x+6(0\leqq x\leqq a)$について，$f(x)$の最小値を求めよ．

(2) $a>0$とする．関数$f(x)=-x^2+6x-8(0\leqq x\leqq a)$について，$f(x)$の最小値を求めよ．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.2.4の解答（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(1)

$f(x)=x^2-4x+6=(x-2)^2+2$ $y=f(x)$のグラフは下に凸の放物線で，軸は直線$x=2$

(i) $0<a<2$のとき

グラフは右の図のようになる．

$x=a$のとき最小となり，最小値$f(a)=a^2-4a+6$

(ii) $2\leqq a$のとき

グラフは右の図のようになる．

$x=2$のとき最小となり，最小値$f(2)=2$

よって，(i)，(ii)より，

\[
\{\begin{array}{ll}0<a<2\text{のとき，}&x=a\text{で最小値}a^2-4a+6\\2\leqq a\text{のとき，}&x=2\text{で最小値}2\end{array}
\]

(2)

$f(x)=-x^2+6x-8=-(x-3)^2+1$ $y=f(x)$のグラフは上に凸の放物線で，軸は直線$x=3$．

(i) $0<a<6$のとき

グラフは右の図のようになる．

$x=0$のとき最小となり，最小値$f(0)=-8$

(ii) $a=6$のとき

グラフは右の図のようになる．

$x=0,6$のとき最小となり，最小値$f(0)=f(6)=-8$

(iii) $6<a$のとき

グラフは右の図のようになる．

$x=a$のとき最小となり，最小値$f(a)=-a^2+6a-8$

よって，(i)〜(iii)より，

\[
\{\begin{array}{ll}0<a<6\text{のとき，}&x=0\text{で最小値}-8\\a=6\text{のとき，}&x=0,6\text{で最小値}-8\\6<a\text{のとき，}&x=a\text{で最小値}-a^2+6a-8\end{array}
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.2.4の側注：定義域が拡大するときの最大・最小（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
定義域$0\leqq x\leqq a$の中央である$x=\frac{a}{2}$と軸の直線$x=2$が一致するときを境にして，場合分けをする．ここでは，$\frac{a}{2}=2$，すなわち，$a=4$を境に場合分けをする．

% 側注2
定義域の両端における$y$座標が等しくなる．

% 側注3
$x=a$の方が軸から遠いので，$x=a$で最大となる．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.2.4の側注：定義域が拡大するときの最大・最小（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
定義域$0\leqq x\leqq a$に軸が含まれるとき，最小となる点は頂点となるので，軸を含むか否かで場合分けをする．

% 側注2
$x=0$の方が軸から遠いので，$x=0$で最小となる．

% 側注3
定義域の両端における$y$座標が等しくなる．

% 側注4
$x=a$の方が軸から遠いので，$x=a$で最小となる．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.2.4の考え方・補足など：定義域が拡大するときの最大・最小（One More）★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 考え方
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

$y=f(x)$のグラフは下に凸の放物線で，軸から遠いほど$y$の値は大きくなる．また，定義域は$a$の値が大きくなるにつれて拡大する．定義域の拡大に伴い，最大値をとる端点も変わるので，軸からの距離を考えて$a$について場合分けをする．ここでは，定義域の中央と軸が一致するときに注目する．

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 注意
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

下の問題の(1)の結果と合わせると，最大値と最小値は次のようにまとめることができる．

(i) $0<a<2$

(ii) $2\leqq a<4$

(iii) $a=4$

(iv) $4<a$