【数学I】例題3.3.35：2つの放物線の大小関係2（One More）★★★★

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.3.35：2つの放物線の大小関係2（One More）★★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

2つの2次関数$f(x)=x^2+2x-2,g(x)=-x^2+2x+a+1$について，次の条件を満たすような定数$a$の値の範囲をそれぞれ求めよ．

(1) $-2\leqq x\leqq 2$を満たすすべての実数$x_1,x_2$に対して，$f(x_1)<g(x_2)$

(2) $-2\leqq x\leqq 2$を満たすある実数$x_1,x_2$に対して，$f(x_1)<g(x_2)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.3.35の解答（One More）★★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\[
f(x)=x^2+2x-2=(x+1)^2-3,
\]

\[
g(x)=-x^2+2x+a+1=-(x-1)^2+a+2
\]

(1) $-2\leqq x\leqq 2$を満たすすべての実数$x_1,x_2$に対して$f(x_1)<g(x_2)$が成り立つ条件は，$-2\leqq x\leqq 2$において，「$(f(x)$の最大値)$<(g(x)$の最小値)」が成り立つときである．

$-2\leqq x\leqq 2$において，$f(x)$の最大値は，$f(2)=6$，$g(x)$の最小値は，$g(-2)=a-7$

したがって，$a-7>6$

よって，$a>13$

(2) $-2\leqq x\leqq 2$を満たすある実数$x_1,x_2$に対して$f(x_1)<g(x_2)$が成り立つ条件は，$-2\leqq x\leqq 2$において，「$(f(x)$の最小値)$<(g(x)$の最大値)」が成り立つときである．

$-2\leqq x\leqq 2$において，$f(x)$の最小値は，$f(-1)=-3$，$g(x)$の最大値は，$g(1)=a+2$

したがって，$a+2>-3$

よって，$a>-5$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.3.35：2つの放物線の大小関係2（One More）★★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

2つの2次関数$f(x)=x^2-4x+5,g(x)=-x^2+a-3$について，次の条件を満たすような定数$a$の値の範囲をそれぞれ求めよ．

(1) $-1\leqq x\leqq 3$を満たすすべての実数$x_1,x_2$に対して，$f(x_1)<g(x_2)$

(2) $-1\leqq x\leqq 3$を満たすある実数$x_1,x_2$に対して，$f(x_1)<g(x_2)$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.3.35の解答（One More）★★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\[
f(x)=x^2-4x+5=(x-2)^2+1
\]

(1) $-1\leqq x\leqq 3$を満たすすべての実数$x_1,x_2$に対して$f(x_1)<g(x_2)$が成り立つ条件は，$-1\leqq x\leqq 3$において，「$(f(x)$の最大値)$<(g(x)$の最小値)」が成り立つときである．

$-1\leqq x\leqq 3$において，$f(x)$の最大値は，$f(-1)=10$，$g(x)$の最小値は，$g(3)=a-12$

したがって，$a-12>10$

よって，$a>22$

(2) $-1\leqq x\leqq 3$を満たすある実数$x_1,x_2$に対して$f(x_1)<g(x_2)$が成り立つ条件は，$-1\leqq x\leqq 3$において，「$(f(x)$の最小値)$<(g(x)$の最大値)」が成り立つときである．

$-1\leqq x\leqq 3$において，$f(x)$の最小値は，$f(2)=1$，$g(x)$の最大値は，$g(0)=a-3$

したがって，$a-3>1$

よって，$a>4$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.3.35の側注：2つの放物線の大小関係2（One More）★★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$f(2)=(2+1)^2-3=6$，$g(-2)=-(-2-1)^2+a+2=a-7$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 問題I3.3.35の側注：2つの放物線の大小関係2（One More）★★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 側注1
$f(-1)=(-1-2)^2+1=10$，$g(3)=-3^2+a-3=a-12$

% 側注2
$f(2)=(2-2)^2+1=1$，$g(0)=0^2+a-3=a-3$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 例題I3.3.35の考え方・補足など：2つの放物線の大小関係2（One More）★★★★
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

「すべての（ある）実数$x$に対して$f(x)<g(x)$」と「すべての（ある）実数$x_1,x_2$に対して$f(x_1)<g(x_2)$」は意味が異なる．「すべての（ある）実数$x$」は，$f(x)$と$g(x)$に同じ$x$の値を代入したときの大小関係を考えているのに対し，「すべての（ある）実数$x_1,x_2$」は，$f(x)$と$g(x)$に異なる$x$の値を代入してもよいことに注意すること．なお，「すべての（ある）実数$x_1,x_2$」の場合は，異なる$x$の値を考えるので，$F(x)=f(x)-g(x)$とまとめることはできない．

(1) 「すべての実数$x_1,x_2$に対して$f(x_1)<g(x_2)$」は，異なる$x_1,x_2$をとったときも，点$(x_1,f(x_1))$は常に点$(x_2,g(x_2))$の下側にある．すなわち，$(f(x)$の最大値)$<(g(x)$の最小値)が成り立つ．

(2) 「ある実数$x_1,x_2$に対して$f(x_1)<g(x_2)$」は，ある$x_1,x_2$をとると，点$(x_1,f(x_1))$が点$(x_2,g(x_2))$の下側にある．すなわち，$(f(x)$の最小値)$<(g(x)$の最大値)が成り立つ．

(1)

(2)