Cách khác:
Đk: \(x\ge2,y\ge3\)
Với a,b\(\ge\) 0có:
\(a+b\le\sqrt{2\left(a^2+b^2\right)}\) <=> \(a^2+2ab+b^2\le2a^2+2b^2\) <=> \(0\le a^2-2ab+b^2\)
<=>\(0\le\left(a-b\right)^2\)
Dấu "=" xảy ra <=>a=b>0
Áp dụng bđt trên có:
\(S=\sqrt{x-2}+\sqrt{y-3}\le\sqrt{2\left(x-2+y-3\right)}=\sqrt{2\left(6-2-3\right)}\)(do x+y=6)
=> \(S\le\sqrt{2}\)
Dấu "=" xảy ra <=> \(\left\{{}\begin{matrix}\sqrt{x-2}=\sqrt{y-3}\\x+y=6\end{matrix}\right.\)
<=> \(\left\{{}\begin{matrix}x-2=y-3\\x+y=6\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}y-x=1\\x+y=6\end{matrix}\right.\) <=> x=2,5 và y=3,5(t/m)
\(S^2\le\left(1+1\right)\left(x-2+y-3\right)=2\left(x+y-5\right)=2\)
\(\Rightarrow S\le\sqrt{2}\)
\(\Rightarrow S_{max}=\sqrt{2}\) khi \(\left\{{}\begin{matrix}x-2=y-3\\x+y=6\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=\frac{5}{2}\\y=\frac{7}{2}\end{matrix}\right.\)
ĐKXĐ :\(x\ge2;y\ge3\)
\(S^2=x-2+y-3+2\sqrt{\left(x-2\right)\left(y-3\right)}\)
\(\ge6-2-3+0=1\)
Vì \(S>0\Rightarrow S\ge1\)
Vậy \(Max_S=1\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=2\\y=4\end{matrix}\right.\\\left\{{}\begin{matrix}x=3\\y=3\end{matrix}\right.\end{matrix}\right.\)
