Tìm `A_{max}` biết `A = (-2|x-2018| – 2021)/(2020+|x – 2018|)`

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1. Đáp án:

   $\begin{array}{l}
   A = \dfrac{{ – 2\left| {x – 2018} \right| – 2021}}{{2020 + \left| {x – 2018} \right|}}\\
    = \dfrac{{ – 2\left| {x – 2018} \right| – 4040 + 2019}}{{2020 + \left| {x – 2018} \right|}}\\
    = \dfrac{{ – 2.\left( {2020 + \left| {x – 2018} \right|} \right) + 2019}}{{2020 + \left| {x – 2018} \right|}}\\
    =  – 2 + \dfrac{{2019}}{{2020 + \left| {x – 2018} \right|}}\\
   Do:\left| {x – 2018} \right| \ge 0\\
    \Leftrightarrow \left| {x – 2018} \right| + 2020 \ge 2020\\
    \Leftrightarrow \dfrac{1}{{2020 + \left| {x – 2018} \right|}} \le \dfrac{1}{{2020}}\\
    \Leftrightarrow \dfrac{{2019}}{{2020 + \left| {x – 2018} \right|}} \le \dfrac{{2019}}{{2020}}\\
    \Leftrightarrow  – 2 + \dfrac{{2019}}{{2020 + \left| {x – 2018} \right|}} \le  – 2 + \dfrac{{2019}}{{2020}}\\
    \Leftrightarrow A \le \dfrac{{ – 2021}}{{2020}}\\
    \Leftrightarrow GTLN:A = \dfrac{{ – 2021}}{{2020}}\\
   Khi:x = 2018
   \end{array}$

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