Question

Vs x,y thỏa mãn ; x-\(\sqrt{x+6}=\sqrt{y+6}\) -y

Tìm GTLN,GTNN của P= x+y

Answer 1

\(x-\sqrt{x+6}=\sqrt{y+6}-y\)

\(\Leftrightarrow x+y=\sqrt{x+6}+\sqrt{y+6}\)

Áp dụng BĐT Bu nhi a cốp xki ta có :

\(\left(x+y\right)^2=\left(\sqrt{x+6}+\sqrt{y+6}\right)^2\le2\left(x+y+12\right)\)

\(\Leftrightarrow\left(x+y\right)^2-2\left(x+y\right)-24\le0\)

\(\Leftrightarrow\left(x+y+4\right)\left(x+y-6\right)\le0\)

\(\Leftrightarrow-4\le x+y\le6\)

Vậy \(MIN_P=-4\) khi \(x=y=-2\) ; \(MAX_P=6\) khi \(x=y=3\)