已知x,y为实数,比较X²+y²也与3x+y-3的大小

已知x,y为实数,比较X²+y²也与3x+y-3的大小
已知x,y为实数,比较X²+y²也与3x+y-3的大小

已知x,y为实数,比较X²+y²也与3x+y-3的大小
x^2+y^2-(3x+y-3)
=(x^2-3x+9/4)+(y^2-y+1/4)+1/2
=(x-3/2)^2+(y-1/2)^2+1/2
≥1/2
因此
x^2+y^2>3x+y-3

X²+y²-(3x+y-3)
=X²+y²-3x+y+9/4+1/4+2/4
=(x-3/2)²+(y-1/2)²+1/2>0
∴X²+y²>3x+y-3

因为(x²+y²)-(3x+y-3)=x²-3x+y²-y+3
=(x²-3x+9/4)+(y²-y+1/4)+1/2
=(x-3/2)²+(y-1/2)²+1/2
>0
所以x²+y²>3x+y-3