Đáp án:
\({{2 - 3x} \over x} + {{2 - 3y} \over y} + {{2 - 3z} \over z} \ge 0\)
\(Dau\,\,'' = ''\,\,xay\,\,ra \Leftrightarrow x = y = z = {2 \over 3}.\)
Giải thích các bước giải:
\(\eqalign{
& {{2 - 3x} \over x} + {{2 - 3y} \over y} + {{2 - 3z} \over z} \cr
& = {2 \over x} - 3 + {2 \over y} - 3 + {2 \over z} - 3 \cr
& = 2\left( {{1 \over x} + {1 \over y} + {1 \over z}} \right) - 9 \cr
& Ta\,\,co:\,\,\left( {x + y + z} \right)\left( {{1 \over x} + {1 \over y} + {1 \over z}} \right) \ge 9\,\,\left( {Co\, - si} \right) \cr
& \Rightarrow 2\left( {{1 \over x} + {1 \over y} + {1 \over z}} \right) \ge 9 \cr
& \Rightarrow 2\left( {{1 \over x} + {1 \over y} + {1 \over z}} \right) - 9 \ge 0 \cr
& vay\,\,{{2 - 3x} \over x} + {{2 - 3y} \over y} + {{2 - 3z} \over z} \ge 0 \cr
& Dau\,\,'' = ''\,\,xay\,\,ra \cr
& \Leftrightarrow \left\{ \matrix{
x = y = z \hfill \cr
x + y + z = 2 \hfill \cr} \right. \Leftrightarrow x = y = z = {2 \over 3}. \cr} \)
