【数学I】例題3.3.34：2つの放物線の大小関係1（One More）★★★★

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% 例題I3.3.34：2つの放物線の大小関係1（One More）★★★★
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2つの2次関数$f(x)=x^2+2ax+9,g(x)=-x^2+6ax-9$について，次の条件を満たすような定数$a$の値の範囲を求めよ．

(1) すべての実数$x$に対して$f(x)>g(x)$

(2) ある実数$x$に対して$f(x)<g(x)$

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% 例題I3.3.34の解答（One More）★★★★
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$F(x)=f(x)-g(x)$とすると，

\[
F(x)=2x^2-4ax+18=2(x-a)^2-2a^2+18
\]

(1) すべての実数$x$に対して$f(x)>g(x)$が成り立つ条件は，すべての実数$x$に対して$F(x)>0$，すなわち，$(F(x)$の最小値$)>0$が成り立つことと同じである．

$F(x)$は$x=a$で最小値$-2a^2+18$をとるから，$-2a^2+18>0$

したがって，$(a+3)(a-3)<0$

よって，$-3<a<3$

(2) ある実数$x$に対して$f(x)<g(x)$が成り立つ条件は，ある実数$x$に対して$F(x)<0$，すなわち，$(F(x)$の最小値$)<0$が成り立つことと同じである．

したがって，$-2a^2+18<0$

ゆえに，$(a+3)(a-3)>0$

よって，$a<-3,3<a$

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% 問題I3.3.34：2つの放物線の大小関係1（One More）★★★★
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2つの2次関数$f(x)=x^2+3ax+20,g(x)=-x^2+7ax-15$について，次の条件を満たすような定数$a$の値の範囲を求めよ．

(1) すべての実数$x$に対して$f(x)>g(x)$

(2) ある実数$x$に対して$f(x)<g(x)$

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% 問題I3.3.34の解答（One More）★★★★
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$F(x)=f(x)-g(x)$とすると，

\[
F(x)=2x^2-4ax+35=2(x-a)^2-2a^2+35
\]

(1) すべての実数$x$に対して$f(x)>g(x)$が成り立つ条件は，すべての実数$x$に対して$F(x)>0$，すなわち，$(F(x)$の最小値$)>0$が成り立つことと同じである．

$F(x)$は$x=a$で最小値$-2a^2+35$をとるから，$-2a^2+35>0$

したがって，$a^2<\frac{35}{2}$

よって，$-\frac{\sqrt{70}}{2}<a<\frac{\sqrt{70}}{2}$

(2) ある実数$x$に対して$f(x)<g(x)$が成り立つ条件は，ある実数$x$に対して$F(x)<0$，すなわち，$(F(x)$の最小値$)<0$が成り立つことと同じである．

したがって，$-2a^2+35<0$

ゆえに，$a^2>\frac{35}{2}$

よって，$a<-\frac{\sqrt{70}}{2},\frac{\sqrt{70}}{2}<a$

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% 例題I3.3.34の側注：2つの放物線の大小関係1（One More）★★★★
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% 側注1
判別式の符号に注目して，$D<0$を用いてもよい．

% 側注2
判別式の符号に注目して，$D>0$を用いてもよい．

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% 問題I3.3.34の側注：2つの放物線の大小関係1（One More）★★★★
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% 側注1
判別式の符号に注目して，$D<0$を用いてもよい．

% 側注2
判別式の符号に注目して，$D>0$を用いてもよい．

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% 例題I3.3.34の考え方・補足など：2つの放物線の大小関係1（One More）★★★★
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% 考え方
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$y=f(x),y=g(x)$のそれぞれのグラフを考えるのではなく，$F(x)=f(x)-g(x)$とし，$F(x)$の条件を考えるとよい．$F(x)$の条件として，$F(x)$の最小値（判別式$D$の符号でもよい）を考えることで，$a$の範囲が求められる．

(1) 「すべての実数$x$に対して$f(x)>g(x)$」$\cdots(\mathrm{i})$は「すべての実数$x$に対して$F(x)>0$」$\cdots(\mathrm{i})^\prime$と同じである．

(2) 「ある実数$x$に対して$f(x)<g(x)$」$\cdots(\mathrm{ii})$は「ある実数$x$に対して$F(x)<0$」$\cdots(\mathrm{ii})^\prime$と同じである．

$(\mathrm{i})$

$(\mathrm{i})^\prime$ $(\mathrm{ii})$

$(\mathrm{ii})^\prime$