a: \(P=x^2+y^2-6x-2y+17\)
\(=x^2-6x+9+y^2-2y+1+7\)
\(=\left(x-3\right)^2+\left(y-1\right)^2+7\ge7\forall x,y\)
Dấu '=' xảy ra khi x-3=0 và y-1=0
=>x=3 và y=1
b: \(Q=x^2+xy+y^2-3x-3y+999\)
\(=x^2+x\left(y-3\right)+y^2-3y+999\)
\(=x^2+2\cdot x\cdot\left(\frac12y-\frac32\right)+\left(\frac12y-\frac32\right)^2+y^2-3y-\left(\frac12y-\frac32\right)^2+999\)
\(=\left(x+\frac12y-\frac32\right)^2+y^2-3y-\left(\frac14y^2-\frac32y+\frac94\right)+999\)
\(=\left(x+\frac12y-\frac32\right)^2+\frac34y^2-\frac32y-\frac94+999\)
\(=\left(x+\frac12y-\frac32\right)^2+\frac34\left(y^2-2y-3\right)+999\)
\(=\left(x+\frac12y-\frac32\right)^2+\frac34\left(y^2-2y+1-4\right)+999\)
\(=\left(x+\frac12y-\frac32\right)^2+\frac34\left(y-1\right)^2+996\ge996\forall x,y\)
Dấu '=' xảy ra khi \(\begin{cases}y-1=0\\ x+\frac12y-\frac32=0\end{cases}\Rightarrow\begin{cases}y=1\\ x=-\frac12y+\frac32=-\frac12+\frac32=\frac22=1\end{cases}\)
c: \(R=2x^2+2xy_{}+y^2-2x+2y+15\)
\(=x^2-4x+4+x^2+2xy+y^2+2x+2y+11\)
\(=\left(x-2\right)^2+x^2+2xy+y^2+2x+2y+1+10\)
\(=\left(x-2\right)^2+\left(x+y+1\right)^2+10\ge10\forall x,y\)
Dấu '=' xảy ra khi \(\begin{cases}x-2=0\\ x+y+1=0\end{cases}\Rightarrow\begin{cases}x=2\\ y=-x-1=-2-1=-3\end{cases}\)
d: \(S=x^2+26y^2-10xy+14x-76y+59\)
\(=x^2-10xy+25y^2+14x-70y+y^2-6y+59\)
\(=\left(x-5y\right)^2+14\left(x-5y\right)+49+y^2-6y+9+1\)
\(=\left(x-5y+7\right)^2+\left(y-3\right)^2+1\ge1\forall x,y\)
Dấu '=' xảy ra khi \(\begin{cases}y-3=0\\ x-5y+7=0\end{cases}\Rightarrow\begin{cases}y=3\\ x=5y-7=5\cdot3-7=15-7=8\end{cases}\)
e: \(T=x^2-4xy+5y^2+10x-22y+28\)
\(=x^2-4xy+4y^2+10x-20y+y^2-2y+28\)
\(=\left(x-2y\right)^2+10\left(x-2y\right)+25+y^2-2y+1+2\)
\(=\left(x-2y+5\right)^2+\left(y-1\right)^2+2\ge2\forall x,y\)
Dấu '=' xảy ra khi \(\begin{cases}y-1=0\\ x-2y+5=0\end{cases}\Rightarrow\begin{cases}y=1\\ x=2y-5=2\cdot1-5=2-5=-3\end{cases}\)
