\(\begin{array}{l}
1)\,\,\,A = {x^2} - 2x + 5 = {\left( {x - 1} \right)^2} + 4 \ge 4\\
Dau\,\, = \,\,xay\,\,\,ra \Leftrightarrow x - 1 = 0 \Leftrightarrow x = 1.\\
Vay\,\,Min\,\,A = 4\,\,khi\,\,x = 1.\\
2)\,\,\,B = 4x - {x^2} + 3 = - \left( {{x^2} - 4x + 4} \right) + 4 + 3\\
= - {\left( {x - 2} \right)^2} + 7 \le 7\\
Dau\,\, = \,\,xay\,\,ra \Leftrightarrow x - 2 = 0 \Leftrightarrow x = 2\\
Vay\,\,Max\,\,B = 7\,\,khi\,\,\,x = 2.\\
3)\,\,C = {x^2} + {y^2} - x + 6y + 10 = {x^2} - 2.\frac{1}{2}x + \frac{1}{4} + \left( {{y^2} + 6y + 9} \right) + \frac{3}{4}\\
= {\left( {x - \frac{1}{2}} \right)^2} + {\left( {y + 3} \right)^2} + \frac{3}{4} \ge \frac{3}{4}\\
Dau\,\, = \,\,\,xay\,\,ra \Leftrightarrow \left\{ \begin{array}{l}
x - \frac{1}{2} = 0\\
y + 3 = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = \frac{1}{2}\\
y = - 3
\end{array} \right.\\
Vay\,\,Min\,C = \frac{3}{4}\,\,\,khi\,\,\,x = \frac{1}{2};\,\,\,y = - 3.\\
4)\,\,D = 5x - {x^2} = - \left( {{x^2} - 2.\frac{5}{2}x + \frac{{25}}{4}} \right) + \frac{{25}}{4}\\
= - {\left( {x - \frac{5}{2}} \right)^2} + \frac{{25}}{4} \le \frac{{25}}{4}\\
Dau\,\,\, = \,\,\,xay\,\,ra \Leftrightarrow x - \frac{5}{2} = 0 \Leftrightarrow x = \frac{5}{2}\\
Vay\,\,MaxD = \frac{{25}}{4}\,\,\,khi\,\,\,x = \frac{5}{2}.\\
5)\,\,E = {x^2} + 10{y^2} - 6xy - 2y + 3 = {x^2} - 6xy + 9{y^2} + {y^2} - 2y + 1 + 2\\
= {\left( {x - 3y} \right)^2} + {\left( {y - 1} \right)^2} + 2 \ge 2\\
Dau\,\, = \,\,xay\,\,ra\, \Leftrightarrow \left\{ \begin{array}{l}
x - 3y = 0\\
y - 1 = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = 3\\
y = 1
\end{array} \right.\\
Vay\,\,Min\,E = 2\,\,\,khi\,\,x = 3;\,\,y = 1.
\end{array}\)
