Question

当x>1时,求函数f(x)=(x*x-3x+1)/(x+1)的值域

Answer 1

f(x)=(x^2-3x+1)/(x+1) = (x^2+x-4x-4+5)/(x+1) = x - 4 + 5/(x+1)

f'(x) = 1 - 5/(x+1)^2 = { (x+1)^2 - 5 ] / (x+1)^2

x＞1

x∈（1,-1+√5）时,f'(x)＜0,f(x)单调减

x∈（-1+√5,+∞）时,f'(x)＞0,f(x)单调增

x=-1+√5时,最小值f(x)min= -1+√5 - 4 + 5/(-1+√5+1) = -5+2√5

值域【-5+2√5,+∞）