求证:不论x,y为何值,多项式x平方-xy+2x+y+5/2永远大于0

Question

求证:不论x,y为何值,多项式x平方-xy+2x+y+5/2永远大于0

Answer 1

x平方-y+2x+y²+5/2?
=x²+2x+1-1+y²-y+1/4-1/4+5/2
=(x+1)²+(y-1/2)²-5/4+10/4
=(x+1)²+(y-1/2)²+5/4
(x+1)²>=0,(y-1/2)²>=0,
(x+1)²+(y-1/2)²+5/4>=5/4
所以不论x,y为何值,多项式x平方-y+2x+y²+5/2永远大于0.