试比较5x²+y²+z²与2xy+4x+2z-2

试比较5x²+y²+z²与2xy+4x+2z-2
试比较5x²+y²+z²与2xy+4x+2z-2

试比较5x²+y²+z²与2xy+4x+2z-2
5x^2+y^2+z^2-(2xy+4x+2z-2)
=5x^2+y^2+z^2-2xy-4x-2z+2
=(4x^2-4x+1)+(x^2-2xy+y^2)+(z^2-2z+1)
=(2x-1)^2+(x-y)^2+(z-1)^2>=0
5x^2+y^2+z^2>=2xy+4x+2z-2

5x²+y²+z²-（2xy+4x+2z-2）
= 5x²+y²+z²-2xy-4x-2z+2
= x²-2xy+y²+4x²-4x+1+z²-2z+1
= (x-y)²+(2x-1)²+(z-1)²>=0
所以5x²+y²+z²>=2xy+4x+2z-2