Lời giải:
Xét biểu thức B:
\(B=x^2+2y^2-2x+2y+2xy+15\)
\(B=(x^2+y^2+1+2xy-2x-2y)+(y^2+4y+4)+10\)
\(B=(x+y-1)^2+(y+2)^2+10\)
Thấy rằng \(\left\{\begin{matrix} (x+y-1)^2\geq 0\\ (y+2)^2\geq 0\end{matrix}\right.\forall x,y\in\mathbb{R}\)
\(\Rightarrow B\geq 10\)
Vậy \(B_{\min}=10\Leftrightarrow \left\{\begin{matrix} x+y-1=0\\ y+2=0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x=3\\ y=-2\end{matrix}\right.\)
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Xét biểu thức C
\(C=x^2+y^2+y+x+y\)
\(C=x^2+y^2+2y+x\)
\(C=(x^2+x+\frac{1}{4})+(y^2+2y+1)-\frac{5}{4}\)
\(C=(x+\frac{1}{2})^2+(y+1)^2-\frac{5}{4}\)
Ta thấy \(\left\{\begin{matrix} (x+\frac{1}{2})^2\geq 0\\ (y+1)^2\geq 0\end{matrix}\right.\forall x,y\in\mathbb{R}\)
\(\Rightarrow C\geq -\frac{5}{4}\) hay \(C_{\min}=\frac{-5}{4}\)
Dấu bằng xảy ra khi \(\left\{\begin{matrix} x+\frac{1}{2}=0\\ y+1=0\end{matrix}\right.\Leftrightarrow \left\{\begin{matrix} x=\frac{-1}{2}\\ y=-1\end{matrix}\right.\)
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Xét biểu thức D
\(D=x^2-2x+y^2-4y+7\)
\(D=(x^2-2x+1)+(y^2-4y+4)+2\)
\(D=(x-1)^2+(y-2)^2+2\)
Thấy rằng \(\left\{\begin{matrix} (x-1)^2\geq 0\\ (y-2)^2\geq 0\end{matrix}\right.\forall x,y\in\mathbb{R}\)
\(\Rightarrow D\geq 2\Leftrightarrow D_{\min}=2\)
Dấu bằng xảy ra khi \(\left\{\begin{matrix} x-1=0\\ y-2=0\end{matrix}\right.\Leftrightarrow x=1; y=2\)
\(C=x^2+y^2+y+x+y\\ =x^2+y^2+2y+x\\ \left(x^2+2.x.\dfrac{1}{2}+\dfrac{1}{4}\right)+\left(y^2+2y+1\right)-\dfrac{5}{4}\\ =\left(x+\dfrac{1}{2}\right)^2+\left(y+1\right)^2-\dfrac{5}{4}\ge-\dfrac{5}{4}\)
Dấu "=" xảy ra khi x=-1/2;y=-1