设f'(x)在(0,+∞)上单调递增,且f(x)=0,证明F(x)=f(x)/x在(0,+∞)上单调增加_数学解答

设f'(x)在(0,+∞)上单调递增,且f(x)=0,证明F(x)=f(x)/x在(0,+∞)上单调增加

人气：476 ℃　时间：2020-06-12 10:07:14

解答

你是说f(0)=0么?
当x1>x2>0的时候,
F(x1)-F(x2)=f(x1)/x1-f(x2)/x2
=f(x1)/x1-f(x2)/x1-(x1-x2)f(x2)/x1x2
=[f(x1)-f(x2)]/x1-(x1-x2)f(x2)/x1x2
=f'(c1)(x1-x2)/x1-(x1-x2)f'(c2)/x1
=(x1-x2)(f'(c1)-f'(c2))/x1>0
(x1>c1>x2>c2>0)
因此单调递增.