\(xy+yz+zx+2xyz=1\)

\(\Leftrightarrow2xy+2yz+2zx+2x+2y+2z+2=xy+yz+zx+2x+2y+2z+3\)

\(\Leftrightarrow2\left(x+1\right)\left(y+1\right)\left(z+1\right)=\left(x+1\right)\left(y+1\right)+\left(y+1\right)\left(z+1\right)+\left(z+1\right)\left(x+1\right)\)

\(\Leftrightarrow\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=2\)

\(\Rightarrow2\ge\frac{9}{x+y+z+3}\Rightarrow x+y+z\ge\frac{3}{2}\)

\(\Rightarrow x^2+y^2+z^2\ge\frac{1}{3}\left(x+y+z\right)^2\ge\frac{3}{4}\)

\(P^2\le3\left[3-\left(x^2+y^2+z^2\right)\right]\le3\left(3-\frac{3}{4}\right)=\frac{27}{4}\)

\(\Rightarrow P\le\frac{3\sqrt{3}}{2}\)

\(P_{max}=\frac{3\sqrt{3}}{2}\) khi \(x=y=z=\frac{1}{2}\)

câu 1

x^2 -5x +y^2+xy -4y +2014 

=(y^2+xy +1/4x^2) -4(y+1/2x)+4 +3/4x^2-3x+2010

=(y+1/2x-2)^2 +3/4(x^2-4x+4)+2007

=(y+1/2x-2)^2 +3/4(x-2)^2 +2007

GTNN là 2007<=> x=2 và y=1

Điều kiện <=> y² =1 -(x-2)² \(\ge0< =>\left(x-2\right)^2\le1< =>-1\le x-2\le1< =>1\le x\le3.\)

 m = x²+y² = x² +1 -(x-2)² = 4x -3

=> 4.1-3 \(\le m\le\)4.3-3 <=> \(1\le m\le9\)

m Min =1 khi x =1; m Max= 9 khi x =3

thanks

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giúp e câu b với ạ

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