a) Ta có: \(\left|x+5\right|\ge0\forall x\)
\(\Rightarrow\left|x+5\right|+2023\ge2023\forall x\)
\(\Rightarrow A\ge2023\forall x\)
Dấu \("="\) xảy ra khi: \(x+5=0\Leftrightarrow x=-5\)
Vậy \(Min_A=2023\) khi \(x=-5\).
b) Ta có: \(\left\{{}\begin{matrix}\left|2x+6\right|\ge0\forall x\\\left|y+3x\right|\ge0\forall x,y\end{matrix}\right.\)
\(\Rightarrow\left|2x+6\right|+\left|y+3x\right|\ge0\forall x,y\)
\(\Rightarrow\left|2x+6\right|+\left|y+3x\right|+25\ge25\forall x,y\)
\(\Rightarrow B\ge25\forall x,y\)
Dấu \("="\) xảy ra khi: \(\left\{{}\begin{matrix}2x+6=0\\y+3x=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x=-6\\y=-3x\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=-6:2=-3\\y=-3\cdot\left(-3\right)=9\end{matrix}\right.\)
Vậy \(Min_B=25\) khi \(x=-3;y=9\).
c) Ta có: \(\left\{{}\begin{matrix}\left|12-3x\right|\ge0\forall x\\\left|-y-4x\right|\ge0\forall x,y\end{matrix}\right.\)
\(\Rightarrow\left|12-3x\right|+\left|-y-4x\right|\ge0\forall x,y\)
\(\Rightarrow\left|12-3x\right|+\left|-y-4x\right|-12\ge-12\forall x,y\)
\(\Rightarrow C\ge-12\forall x,y\)
Dấu \("="\) xảy ra khi: \(\left\{{}\begin{matrix}12-3x=0\\-y-4x=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x=12\\y=-4x\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=12:3=4\\y=-4\cdot4=-16\end{matrix}\right.\)
Vậy \(Min_C=-12\) khi \(x=4;y=-16\).
\(\mathit{Toru}\)
