Bài làm
C = x2 - 4xy + 5y2 + 10x - 22y + 28
C = (x2 - 4xy + 4y2) + (10x - 20y) + (y2 - 2y) + 28
C = (x - 2y)2 + 10(x - 2y) + 25 + (y2 - 2y + 1) + 2
C = (x - 2y)2 + 2.(x - 2y).5 + 52 + (y - 1)2 + 2
C = (x - 2y + 5)2 + (y - 1)2 + 2
Vì (x - 2y + 5)2 \(\ge\) 0
(y - 1)2 \(\ge\) 0
=> (x - 2y + 5)2 + (y - 1)2 \(\ge\) 0
=> (x - 2y + 5)2 + (y - 1)2 + 2 \(\ge\) 2 \(\forall\) x; y.
Dấu " = " xảy ra <=> \(\left\{{}\begin{matrix}\left(x-2y+5\right)^2=0\\\left(y-1\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-2y+5=0\\y-1=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=2y-5\\y=1\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=-3\\y=1\end{matrix}\right.\)
Vậy Min C = 2 <=> x = -3; y = 1
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