Đáp án:
$\begin{array}{l}
1)6x - {x^2} - 5\\
= - \left( {{x^2} - 6x + 9} \right) + 9 - 5\\
= - {\left( {x - 3} \right)^2} + 4 \le 4\\
\Leftrightarrow GTLN = 4\,khi:x = 3\\
3)4x - {x^2} + 3\\
= - \left( {{x^2} - 4x + 4} \right) + 4 + 3\\
= - {\left( {x - 2} \right)^2} + 7 \le 7\\
\Leftrightarrow GTLN = 7\,khi:x = 2\\
5)x - {x^2}\\
= - \left( {{x^2} - x + \dfrac{1}{4}} \right) + \dfrac{1}{4}\\
= - {\left( {x - \dfrac{1}{2}} \right)^2} + \dfrac{1}{4} \le \dfrac{1}{4}\\
\Leftrightarrow GTLN = \dfrac{1}{4}\,khi:x = \dfrac{1}{2}\\
2){x^2} + 5x + 7\\
= {x^2} + 2.x.\dfrac{5}{2} + \dfrac{{25}}{4} + \dfrac{3}{4}\\
= {\left( {x + \dfrac{5}{2}} \right)^2} + \dfrac{3}{4} \ge \dfrac{3}{4}\\
\Leftrightarrow GTNN = \dfrac{3}{4}\,khi:x = - \dfrac{5}{2}\\
4){x^2} - 20x + 101\\
= {x^2} - 20x + 100 + 1\\
= {\left( {x - 10} \right)^2} + 1 \ge 1\\
\Leftrightarrow GTNN = 1\,khi:x = 10\\
6)4{a^2} + 4a + 2\\
= 4{a^2} + 4a + 1 + 1\\
= {\left( {2a + 1} \right)^2} + 1 \ge 1\\
\Leftrightarrow GTNN = 1\,khi:a = - \dfrac{1}{2}\\
8){x^2} - 4xy + 5{y^2} + 10x - 22y + 28\\
= {x^2} + 4{y^2} + 25 - 4xy + 10x - 20y\\
+ {y^2} - 2y + 1 + 2\\
= {\left( {x - 2y + 5} \right)^2} + {\left( {y - 1} \right)^2} + 2 \ge 2\\
\Leftrightarrow GTNN = 2\,khi:\left\{ \begin{array}{l}
x = 2y - 5\\
y = 1
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = - 3\\
y = 1
\end{array} \right.\\
10){x^2} + 3x + 7\\
= {x^2} + 2.x.\dfrac{3}{2} + \dfrac{9}{4} + \dfrac{{19}}{4}\\
= {\left( {x + \dfrac{3}{2}} \right)^2} + \dfrac{{19}}{4} \ge \dfrac{{19}}{4}\\
\Leftrightarrow GTNN = \dfrac{{19}}{4}\,khi:x = - \dfrac{3}{2}
\end{array}$
