Đáp án:
\(\left\{ \begin{array}{l}
\mathop {\min }\limits_{\left[ { - 3;\,\,4} \right]} y = - 2\,\,\,khi\,\,\,x = 2\\
\mathop {\max }\limits_{\left[ { - 3;\,\,4} \right]} y = 4\,\,\,khi\,\,\,x = 4
\end{array} \right..\)
Giải thích các bước giải:
\(\begin{array}{l}
y = 3\left| {x - 2} \right| - \left| {2x - 6} \right|\\
+ )\,\,\,Voi\,\,\,\,x < 2 \Rightarrow \left\{ \begin{array}{l}
\left| {x - 2} \right| = - x + 2\\
\left| {2x - 6} \right| = - 2x + 6
\end{array} \right.\\
\Rightarrow y = 3\left( { - x + 2} \right) - \left( { - 2x + 6} \right) = - x\\
+ )\,\,Voi\,\,\,\,2 \le x < 3 \Rightarrow \left\{ \begin{array}{l}
\left| {x - 2} \right| = x - 2\\
\left| {2x - 6} \right| = - 2x + 6
\end{array} \right.\\
\Rightarrow y = 3\left( {x - 2} \right) - \left( { - 2x + 6} \right) = 5x - 12\\
+ )\,\,Voi\,\,\,\,x \ge 3 \Rightarrow \left\{ \begin{array}{l}
\left| {x - 2} \right| = x - 2\\
\left| {2x - 6} \right| = 2x - 6
\end{array} \right.\\
\Rightarrow y = 3\left( {x - 2} \right) - \left( {2x - 6} \right) = x\\
\Rightarrow y = 3\left| {x - 2} \right| - \left| {2x - 6} \right| = \left\{ \begin{array}{l}
- x\,\,\,\,khi\,\,\,x < 2\\
5x - 12\,\,\,\,khi\,\,\,2 \le x < 3\\
2x - 6\,\,\,khi\,\,\,x \ge 3
\end{array} \right..
\end{array}\)
Ta có đồ thị như trên hình vẽ.
Dựa vào đồ thị hàm số ta thấy: \(\left\{ \begin{array}{l}
\mathop {\min }\limits_{\left[ { - 3;\,\,4} \right]} y = - 2\,\,\,khi\,\,\,x = 2\\
\mathop {\max }\limits_{\left[ { - 3;\,\,4} \right]} y = 4\,\,\,khi\,\,\,x = 4
\end{array} \right..\)
