Question

[Nguyen Thu Hang]

CMR : 

( x² + 5y² - 4xy + 2x - 10y + 14 ) > 0 với mọi x , y

Giúp mình với !!!!!!!

Answer 1

[Lê Tài Bảo Châu]

\(x^2+5y^2-4xy+2x-10y+14\)

\(=\left(x^2+4y^2-4xy+2x-4y+1\right)+\left(y^2-6y+9\right)+4\)

\(=\left(x-2y+1\right)^2+\left(y-3\right)^2+4\)

Vì \(\hept{\begin{cases}\left(x-2y+1\right)^2\ge0;\forall x,y\\\left(y-3\right)^2\ge0;\forall x,y\end{cases}}\)

\(\Rightarrow\left(x-2y+1\right)^2+\left(y-3\right)^2\ge0;\forall x,y\)

\(\Rightarrow\left(x-2y+1\right)^2+\left(y-3\right)^2+4\ge4>0;\forall x,y\)

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