◆値域の問題(つづき)

\begin{align*}
\Longrightarrow~&k=\frac{1}{4} \lor \exists x \exists y \begin{cases} x^2+y^2=k \\ k \neq 0 \land y\neq 0 \\ \exists t \left[ (4k-1)t^2-8kt+(10k-1)=0 \right] \land \exists t \left [\frac{x}{y}=t \right]\end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \exists x \exists y \begin{cases} x^2+y^2=k \\ k \neq 0 \land y\neq 0 \\ \exists t \left[ (4k-1)t^2-8kt+(10k-1)=0 \right]\end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \exists x \exists y \begin{cases} x^2+y^2=k \\ k \neq 0 \land y\neq 0 \\ \exists t \left[ (4k-1)t^2-8kt+(10k-1)=0 \land (k=\frac{1}{4} \lor k\neq \frac{1}{4})\right]\end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \exists x \exists y \begin{cases} x^2+y^2=k \\ k \neq 0 \land y\neq 0 \\ \exists t \left[ (4k-1)t^2-8kt+(10k-1)=0 \land k=\frac{1}{4}\right] \\
\lor \exists t \left[(4k-1)t^2-8kt+(10k-1)=0 \land k\neq \frac{1}{4}\right]\end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \exists x \exists y \begin{cases} x^2+y^2=k \\ k \neq 0 \land y\neq 0 \\ \exists t \left[ t=\frac{3}{4} \land k=\frac{1}{4}\right] \lor \exists t \left[\frac{1}{12} \leq k \leq \frac{1}{2} \land k\neq \frac{1}{4}\right]\end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \exists x \exists y \begin{cases} x^2+y^2=k \\ k \neq 0 \land y\neq 0 \\ k=\frac{1}{4} \lor \left( \frac{1}{12} \leq k \leq \frac{1}{2} \land k\neq \frac{1}{4}\right)\end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \begin{cases} \exists x \exists y ( x^2+y^2=k \land y\neq 0 ) \\ k \neq 0 \\ \frac{1}{12} \leq k \leq \frac{1}{2} \end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \begin{cases} \exists x \exists y ( x^2+y^2=k \land y\neq 0 ) \\ \frac{1}{12} \leq k \leq \frac{1}{2} \end{cases}\\
\Longleftrightarrow~&k=\frac{1}{4} \lor \left( k \geq 0 \land \frac{1}{12} \leq k \leq \frac{1}{2} \right)\\
\Longleftrightarrow~&\left( k=\frac{1}{4} \lor k \geq 0 \right) \land \left( k=\frac{1}{4} \lor \frac{1}{12} \leq k \leq \frac{1}{2} \right)\\
\Longleftrightarrow~&k \geq 0 \land \frac{1}{12} \leq k \leq \frac{1}{2}\\
\Longleftrightarrow~&\frac{1}{12} \leq k \leq \frac{1}{2}
\end{align*}

逆に,\(k=\frac{1}{12}\)のとき,前記事\((\ast)\)が成り立つかを調べる.
\begin{align*}
&\exists x \exists y \begin{cases} x^2+y^2=k \\ k \neq 0 \land y\neq 0 \\ \exists t \left[ (4k-1)t^2-8kt+(10k-1)=0\land \frac{x}{y}=t \right]\end{cases}\tag{\(\ast\)}\\
\Longleftrightarrow~&\exists x \exists y \begin{cases} x^2+y^2=\frac{1}{12} \\ \frac{1}{12} \neq 0 \land y\neq 0 \\ \exists t \left[ 4t^2+4t+1=0\land \frac{x}{y}=t \right]\end{cases}\\
\Longleftrightarrow~&\exists x \exists y \begin{cases} x^2+y^2=\frac{1}{12} \\ y\neq 0 \\ \exists t \left[ t=-\frac{1}{2}\right] \land y=-2x \end{cases}\\
\Longleftrightarrow~&\exists x \exists y \begin{cases} x^2+y^2=\frac{1}{12} \\ y=-2x \\ y\neq 0 \end{cases}
\end{align*}
この命題は明らかに真である.

\(k=\frac{1}{2}\)のときも同様に\((\ast)\)は真となる.したがって最大値は\(\frac{1}{2}\),最小値は\(\frac{1}{12}\).

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