一头迷惘的猪
幼苗
共回答了20个问题采纳率:80% 举报
(1)
f'(x) = a - b/x²
f(1) = a + b + c
f'(1) = a - b
切线斜率 = 1, a - b = 1, b = a - 1
切线上, x = 1, y = 0, f(1) = a + b + c = 0, c = -a - b = -a -a + 1 = 1 - 2a
(2)
g(x) = f(x) - lnx
= ax + (a-1)/x + 1 - 2a - lnx
g'(x) = a - (a-1)/x² - 1/x = (ax² - x -a + 1)/x²
= (x -1)[ax - (1 -a)] = 0
x = 1或x = (1-a)/a
a >0, h(x) =(ax² - x -a + 1)为开口向上的抛物线, 与x轴交于A(1, 0), B((1-a)/a, 0), 且在A,B外侧, g'(x) >0
若f(x)大于等于Lnx在[1,+无穷大)上恒成立, 只需g(1) ≥ 0且g(x)在[1, +∞)上是增函数即可.
要做到这一点,须B在A左侧或重合: (1-a)/a ≤ 1, a ≥ 1/2
1年前
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