Question

已知X,Y属于R,求证X平方减去XY+Y平方大于等于X+Y减去1

Answer 1

2(x²-xy+y²)-2(x+y-1)

=(x²-2x+1)+(y²-2y+1)+(x²-2xy+y²)

=(x-1)²+(y-1)²+(x-y)²

平方大于等于0

所以(x-1)²+(y-1)²+(x-y)²>=0

所以2(x²-xy+y²)-2(x+y-1)>=0

所以2(x²-xy+y²)>=2(x+y-1)

所以x²-xy+y²>=x+y-1