Question

với x=a,y=b thì biểu thức A=x^2 + xy + y^2 - 3x + 3y đạt gtnn khi đó 2a-b nhận giá trị bằng

 

Answer 1

\(A=\left(x^2+xy+\dfrac{1}{4}y^2\right)-3\left(x+\dfrac{1}{2}y\right)+\dfrac{9}{4}+\left(\dfrac{3}{4}y^2+\dfrac{9}{2}y\right)-\dfrac{9}{4}\\ A=\left[\left(x+\dfrac{1}{2}y\right)^2-3\left(x+\dfrac{1}{2}y\right)+\dfrac{9}{4}\right]+\dfrac{3}{4}\left(y^2+6y+9\right)-9\\ A=\left(x+\dfrac{1}{2}y-\dfrac{3}{2}\right)^2+\dfrac{3}{4}\left(y+3\right)^2-9\ge9\\ A_{min}=9\Leftrightarrow\left\{{}\begin{matrix}x+\dfrac{1}{2}y=\dfrac{3}{2}\\y=-3\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=-3\end{matrix}\right.\Leftrightarrow2a-b=2\cdot3-3\left(-3\right)=12\)