Question

Tìm GTNN của biểu thức Q=\(\left(\left|x-3\right|+2\right)^2+\left|y+3\right|+2007\)

Answer 1

Ta có: \(\left\{{}\begin{matrix}\left|x-3\right|\ge0\forall x\\\left|y+3\right|\ge0\forall y\end{matrix}\right.\)

+) \(\left|x-3\right|\ge0\Rightarrow\left(\left|x-3\right|+2\right)\ge2\)

\(\Rightarrow\left(\left|x-3\right|+2\right)^2\ge4\)

Dấu ''='' xảy ra \(\Leftrightarrow x=3\)

=> \(Min_{\left(\left|x-3\right|+2\right)^2}=4\Leftrightarrow x=3\)

+) \(\left|y+3\right|\ge0\)

Dấu ''='' xảy ra \(\Leftrightarrow y=-3\)

=> \(Min_{\left|y+3\right|}=0\Leftrightarrow y=-3\)

\(\Rightarrow MIN_Q=4+0+2017=2021\)

Vậy \(MIN_Q=2021\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=-3\end{matrix}\right.\)