Áp dụng BĐT AM-GM ta có:
\(x^2\cdot\dfrac{4}{9}+y^2\cdot\dfrac{4}{9}\ge\dfrac{8xy}{9}\)
\(x^2\cdot\left(\dfrac{4}{3}\right)^2+z^2\cdot\left(\dfrac{1}{3}\right)^2\ge\dfrac{8xz}{9}\)
\(y^2\cdot\left(\dfrac{4}{3}\right)^2+z^2\cdot\left(\dfrac{1}{3}\right)^2\ge\dfrac{8yz}{9}\)
CỘng theo vế 3 BĐt trên ta có:
\(\dfrac{2}{9}\left(10x^2+10y^2+z^2\right)\ge\dfrac{8\left(xy+yz+xz\right)}{9}\)
\(\Leftrightarrow10x^2+10y^2+z^2\ge4\left(xy+yz+xz\right)=4\)
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