Hi there!
In the image attached, we realize point (a,b) is in the given shaded region if, and only if,
\(\left\{ \begin{gathered}
\,0 < a < 4\,\,\,\left( {\text{I}} \right) \hfill \\
\,{a^2} - 4a < b < 0\,\,\,\left( {{\text{II}}} \right) \hfill \\
\end{gathered} \right. \)
\(\left( {\text{I}} \right)\,\,0 < a < 4\,\,\, \) Although \( \,b < 0\, \) is given, we still can have a "yes" and a "no" possibilities:
\( \left. \begin{gathered}
\left( {a,b} \right) = \left( {1, - 2} \right)\,\,\,\,;\,\,\,{a^2} - 4a\mathop < \limits^? b\,\,\,:{\text{yes}}\,\, \hfill \\
\left( {a,b} \right) = \left( {1, - 3} \right)\,\,\,\,;\,\,\,{a^2} - 4a\mathop < \limits^? b\,\,\,:{\text{no}}\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,{\text{INS}}{\text{.}} \)
\( \left( {{\text{II}}} \right)\,\,{a^2} - 4a < b < 0\,\,\, \)
\(\left. {{a^2} - 4a < b < 0\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\,\left( {{\text{II}}} \right)\,\,\,{\text{immediately}} \hfill \\
\,a\left( {a - 4} \right) = {a^2} - 4a < 0\,\,\,\, \Rightarrow \,\,\,\,0 < a < 4\,\,\, \Rightarrow \,\,\,\,\left( {\text{I}} \right)\,\,\,\,\, \hfill \\
\end{gathered} \right.} \right\}\,\,\,\,{\text{SUF}}{\text{.}}\)
Regards,
Fabio.
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Fabio Skilnik ::
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