Question

Tìm GTNN của biểu thức:

A=(|x−3|+2)2+|y+3|+2018

Answer 1

Ta có : \(A=\left(|x-3|+2\right).2+|y+3|+2018\)

                \(=2.|x-3|+4+|y-3|+2018\)

                \(=\left(2.|x-3|+|y+3|\right)+\left(4+2018\right)\)

                \(=\left(2.|x-3|+|y+3|\right)+2022\)

Vì \(|x-3|\ge0\)\(\forall x\)

     \(|y+3|\ge0\)\(\forall y\)

\(\Rightarrow2.|x-3|+|y+3|\ge0\)\(\forall x,y\)

\(\Rightarrow2.|x-3|+|y+3|+2022\ge2022\)\(\forall x,y\)

hay \(A\ge2022\)

\(\Rightarrow minA=2022\Leftrightarrow x-3=0\)và \(y+3=0\)

                                   \(\Leftrightarrow x=3\)và \(y=-3\)

Vậy \(minA=2022\Leftrightarrow x=3\)và \(y=-3\)