Cho x,y>0 thoa man 2(x-y)^2 + (x+1)(y+1) + 7xy = 1 +4.sqrt(xy). Tim GTNN cua P=(3y+1)/sqrt(x) + (3x+1)/sqrt(y) + x + y
$\eqalign{
& P = {{3y + 1} \over {\sqrt x }} + {{3x + 1} \over {\sqrt y }} + x + y = {{3y} \over {\sqrt x }} + {{3x} \over {\sqrt y }} + {3 \over {4\sqrt x }} + {3 \over {4\sqrt y }} + {1 \over {8\sqrt x }} + {1 \over {8\sqrt x }} + {1 \over {8\sqrt y }} + {1 \over {8\sqrt y }} + x + y \cr
& \cos i: \cr
& {{3y} \over {\sqrt x }} + {{3x} \over {\sqrt y }} + {3 \over {4\sqrt x }} + {3 \over {4\sqrt y }} \ge 4\root 4 \of {{{{3^4}xy} \over {{4^2}xy}}} = 6 \cr
& {1 \over {8\sqrt x }} + {1 \over {8\sqrt x }} + {1 \over {8\sqrt y }} + {1 \over {8\sqrt y }} + x + y \ge 6\root 6 \of {{{xy} \over {{8^4}xy}}} = {3 \over 2} \cr
& \to P \ge 6 + {3 \over 2} = {{15} \over 2} \cr
& dau = \leftrightarrow x = y = {1 \over 4} \cr
& thay\;x = y = {1 \over 4}\;vao\;dieu\;kien \to thoa\;man \cr
& vay\;minP = {{15} \over 2} \cr
& \cr} $
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