数学问题。关于导函数的

(1)f(x) = x - x^2
f'(x) = 1 - 2x

0 < x < 1/2, f'(x) > 0。f(x)单调递增。f(x) > f(0) = 0
1/2 < x < 1, f'(x) < 0. f(x)单调递减。f(x) > f(1) = 0

所以，0 < x < 1时，总有f(x) = x - x^2 > 0

(2) g(x) = e^x - x - 1
g'(x) = e^x - 1,
x < 0, g'(x) < 0. g(x)单调递减。g(x) > g(0) = 0.
x > 0, g'(x) > 0. g(x)单调递增。g(x) > g(0) = 0.

所以，当x不等于0时，总有g(x) = e^x - x - 1 > 0, e^x > 1 + x.

(3) h(x) = x - lnx, f(x) = e^x - x, x > 0.
h'(x) = 1 - 1/x, f'(x) = e^x - 1 > 0. f(x)在x>0时单调递增。f(x)>f(0)=1 > 0

0 < x < 1时，h'(x)<0, h(x)单调递减，h(x)>h(1) = 1 > 0.
1 < x 时，h'(x)>0, h(x)单调递增。h(x)>h(1) = 1 > 0.

所以，x>0时，总有f(x) = e^x - x > 0, h（x) = x - lnx > 0.
也即，
lnx < x < e^x.