Tìm giá trị nhỏ nhất a, x^2-x) b, 2x^2+y^2+2xy-4x+2y=10 (giải chi tiết

\[\begin{array}{l} a)\,\,{x^2} - x = {x^2} - 2x.\frac{1}{2} + \frac{1}{4} - \frac{1}{4} = {\left( {x - \frac{1}{2}} \right)^2} - \frac{1}{4} \ge - \frac{1}{4}\\ \Rightarrow Min\,\,\left( {{x^2} - x} \right) = - \frac{1}{4}\,\,\,khi\,\,\,x = \frac{1}{2}.\\ b)\,\,\,B = 2{x^2} + {y^2} + 2xy - 4x + 2y + 10\\ = {x^2} + 2xy + {y^2} + 2x + 2y + 1 + {x^2} - 6x + 9\\ = {\left( {x + y + 1} \right)^2} + {\left( {x - 3} \right)^2} \ge 0\\ Dau\,\, = \,\,xay\,\,ra\,\, \Leftrightarrow \left\{ \begin{array}{l} x + y + 1 = 0\\ x - 3 = 0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} x = 3\\ y = - 4 \end{array} \right.\\ \Rightarrow Min\,\,B = 0\,\,khi\,\,\,\left\{ \begin{array}{l} x = 3\\ y = - 4 \end{array} \right.. \end{array}\]