Question

求函数最小值

求函数x²-3x+3/x-2(x＞2)的最小值
简单那点- -.看不懂

Answer 1

原式=[(x-2)(x-1)+1]/(x-2)
=(x-2)(x-1)/(x-2)+1/(x-2)
=x-1+1/(x-2)
=(x-2)+1/(x-2)+1

x>2
所以x-2>0
所以(x-2)+1/(x-2)+1≥2√[(x-2)*1/(x-2)]+1=2+1=3
所以最小值=3

Answer 2

设f(x)=(x²-3x+3)/(x-2)
=x-1+1/(x-2)
=1+x-2+1/(x-2)(x＞2),
由均值不等式，
f(x)>=1+2=3,
当x=3时取等号，
∴f(x)的最小值=3.