Question

求函数y=x^2-3x+6/x-2(x>2)的取值范围

Answer 1

y=(x^2-3x+6)/(x-2) (x>2)

y=(x²-4x+x-2+8)/(x-2)

y=[(x-2)²+(x-2)+8]/(x-2)

令x-2=t,则t>0

y=[(x-2)²+(x-2)+8]/(x-2)

=(t²+t+8)/t

=t+1+8/t

=t+8/t+1 (t>0)

≥2√(t*8/t)+1

=4√2+1

当且仅当t=8/t即t=2√2,x=2+2√2时等号成立

所以y=x^2-3x+6/x-2(x>2)的取值范围是：y≥4√2+1