Question

这道题的正确答案是什么

Answer 1

(1)

a = 1, f(x) = x - lnx, 定义域x > 0

f'(x) = 1 - 1/x = (x - 1)/x = 0
x = 1
0 < x < 1: f'(x) < 0, 减函数
x > 1: f'(x) > 0, 增函数
最小值f(1) = 1

(2)
f'(x) = (1 - a)x + a - 1/x = [(1 - a)x² + ax - 1]/x = [(1 - a)x + 1](x - 1)/x = 0
a ≥ 2, 分子为开口向下的抛物线

x = 1, x = 1/(a - 1)

a ≥ 2, a - 1 ≥ 1, 1/(a - 1)≤ 1
(i) a = 2
二零点重合, f'(x) ≤ 0, 且仅在x = 1时等号成立, 减函数 (定义域x > 0内)

(ii) a > 2
二零点x = 1, x = 1/(a - 1)不重合
0 < x < 1/(a - 1): f'(x) < 0, 减函数
1/(a - 1) < x < 1: f'(x) > 0, 增函数
x > 1: f'(x) < 0, 减函数

(3)
不像开始想得那样容易，有空再看。