x-\(6\sqrt{xy}\)+13y-12\(\sqrt{y}\)+9=0 đkxđ:x,y\(\ge\)0
\(\Leftrightarrow\)( x-\(2\sqrt{x}.3\sqrt{y}\)+9y )+( 4y-\(2.2\sqrt{y}.3\)+9 )=0
\(\Leftrightarrow\)\(\left(\sqrt{x}-3\sqrt{y}\right)^2\)+\(\left(2\sqrt{y}-3\right)^2\)=0
vì\(\left(\sqrt{x}-3\sqrt{y}\right)^2\)\(\ge0v\text{ới}\forall x,y\ge0\)
\(\left(2\sqrt{y}-3\right)^2\ge0\) với \(\forall y\ge\)0
nên để VT=VP khi dấu "=" xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}x-3\sqrt{y}=0\\2\sqrt{y}-3=0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{x}=3\sqrt{y}\\2\sqrt{y}=3\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{y}=\frac{3}{2}\\\sqrt{x}=3\sqrt{y}\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\frac{9}{4}\\x=9y\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}y=\frac{9}{4}\\x=\frac{81}{4}\end{matrix}\right.\)
TL: Luôn đúng
Vậy với \(\left\{{}\begin{matrix}x=\frac{81}{4}\\y=\frac{9}{4}\end{matrix}\right.\)thì thỏa mãn
