Question

Tìm GTNN của các biểu thức sau:

a) A= \(4\left|x+3\right|+\left|4x-5\right|+12\)

b) B= \(\left|x-6\right|+\left|y+1\right|\) biết \(x-y=3\)

Answer 1

\(A=\left|4x+12\right|+\left|5-4x\right|+12\ge\left|4x+12+5-4x\right|+12=29\)

\(\Rightarrow A_{min}=29\) khi \(\left(4x+12\right)\left(5-4x\right)\ge0\Leftrightarrow-3\le x\le\dfrac{5}{4}\)

\(B=\left|x-6\right|+\left|y+1\right|=\left|x-6\right|+\left|-y-1\right|\ge\left|x-6-y-1\right|=\left|3-7\right|=4\)

\(\Rightarrow B_{min}=4\) khi \(\left(x-6\right)\left(-y-1\right)\ge0\) \(\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge6\\y\le-1\end{matrix}\right.\\\left\{{}\begin{matrix}x\le6\\y\ge-1\end{matrix}\right.\end{matrix}\right.\)