Question

Tìm GTNN :

a) A= (x-1)^2 +(y-3)^2

b) B= 2x^2 +y^2 -2xy -2x +3

Answer 1

a) \(A=\left(x-1\right)^2+\left(y-3\right)^2\ge0\) Do \(\left(x-1\right)^2\ge0;\left(y-3\right)^2\ge0\)

Dấu "=" xảy ra khi

\(\Rightarrow\)\(\begin{cases}\left(x-1\right)^2=0\\\left(y-3\right)^2=0\end{cases}\)\(\Rightarrow\begin{cases}x-1=0\\y-3=0\end{cases}\)\(\Rightarrow x=1;y=3\)

Vậy \(minA=0\) khi x=1;y=3

b) \(B=2x^2+y^2-2xy-2x+3=\left(x^2-2xy+y^2\right)+\left(x^2-2x+1\right)+2\)

\(\Rightarrow B=\left(x-y\right)^2+\left(x-1\right)^2+2\ge2\)

Dấu "=" xảy ra khi:

\(\Leftrightarrow\begin{cases}\left(x-y\right)^2=0\\\left(x-1\right)^2=0\end{cases}\)\(\Rightarrow\begin{cases}x=y\\x=1\end{cases}\)

Vậy minB =2 khi x=y=1