Question

已知函数f(x)=e^x-ax（a∈R）1、求函数f(x)的单调区间

2、当x∈[0,正无穷)时,都有f(x)≥0成立,求实数a的取值范围.

Answer 1

1. f'(x) = e^x - a

(1) a ≤ 0时
e^x > 0, f'(x) > 0
f(x)在(-∞, +∞)内单调递增

(2) a > 0时
f'(x) = e^x - a = 0
x = lna
x < lna时, f'(x) < 0, f(x)单调递减
x > lna时, f(x) > 0, f(x)单调递增

2.
(1)a ≤ 0时, f(0) = 1, f(x) ≥0显然成立
(2) a > 0时, 须极值点在y轴左侧(lna < 0, a < 1)，且f(0) ≥0
f(0) = 1 > 0显然成立, a < 1结合前提: 0 < a < 1
结合(1)(2): a < 1