Question

求完整答案，需要过程

Answer 1

续(2):

F(x)的定义域为x > 0

(i) a = 0

F(x) = lnx - 2 = 0

x = e²

F(x) = 0在[1, e²]上有解

(ii) a > 0

F(a)为最小值, 为lna - 1

x->0和x ->正无穷时，F(x)均趋近于正无穷，所以须同时满足F(a) = lna - 1 < 0, 0 <a < e及如下2个条件之一：
F(1) = a - 2 ≥ 0, a ≥ 2

或F(e²) = 2 + a/e² - 2 = a/e² ≥ 0, a ≥ 0

结合起来，得0 < a < e

另外a = e时，F(a) =lna - 1= 0时，a = e, 此时F(x)与x轴相切于(e, 0), e也满足，即a > 0时的答案是0 < a ≤ e

(iii) a < 0:

F'(x) 恒为正, F'(x)单调增

x->0, F(x) -> 负无穷

x ->0, F(x) ->正无穷

F(x) = 0总有一个解

现在考虑F(e²)的情况

F(e²) = 2 + a/e² - 2 = a/e², 在a < 0时,恒小于0, 即总在x轴下方, 即在[1, e²]内无解

结合(i)(ii), 结果为0 ≤ a ≤ e(参考下图)