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\(Q=\frac{3x+3y+2z}{\sqrt{6\left(x^2+5\right)}+\sqrt{6\left(y^2+5\right)}+\sqrt{z^2+5}}\)
\(\Leftrightarrow Q=\frac{3x+3y+2z}{\sqrt{6\left(x^2+xy+yz+zx\right)}+\sqrt{6\left(y^2+xy+yz+zx\right)}+\sqrt{z^2+xy+yz+zx}}\)
\(\Leftrightarrow Q=\frac{3x+3y+2z}{\sqrt{3\left(x+y\right).2\left(x+z\right)}+\sqrt{3\left(y+x\right).2\left(y+z\right)}+\sqrt{\left(z+x\right).\left(z+y\right)}}\)
\(\Rightarrow Q\ge\frac{3x+3y+2z}{\frac{3\left(x+y\right)+2\left(x+z\right)}{2}+\frac{3\left(y+x\right)+2\left(y+z\right)}{2}+\frac{\left(z+x\right)+\left(z+y\right)}{2}}\)
\(\Rightarrow Q\ge\frac{3x+3y+2z}{\frac{9x+9y+6z}{2}}=\frac{2}{3}\)
Dấu "=" xảy ra khi \(x=y=1\)và \(z=2\)
\(M\in\Delta\Rightarrow M\left(a;-a-2\right)\)
\(\Rightarrow\overrightarrow{MA}=\left(1-a;a+4\right)\) ; \(\overrightarrow{MB}=\left(-a;a+3\right)\); \(\overrightarrow{MC}=\left(-2-a;a+3\right)\)
\(\Rightarrow\overrightarrow{u}=\overrightarrow{MA}+2\overrightarrow{MB}+\overrightarrow{MC}=\left(-1-4a;4a+13\right)\)
\(\Rightarrow\left|\overrightarrow{u}\right|=\sqrt{\left(-1-4a\right)^2+\left(4a+13\right)^2}\)
\(=\sqrt{32a^2+112a+170}=\sqrt{2\left(4a+7\right)^2+72}\ge\sqrt{72}\)
\(\Rightarrow\left|\overrightarrow{u}\right|_{min}\) khi \(4a+7=0\Rightarrow a=-\frac{7}{4}\Rightarrow M\left(-\frac{7}{4};-\frac{1}{4}\right)\)
bất lợi ,khó khăn,....
bất lợi,bất tiện,khó khăn
hok tốt