设f(x)对任意的实数x1,x2有f(x1+x2)=f(x1)f(x2),且f'(0)=1,证明f'(x)=f(x)
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设f(x)对任意的实数x1,x2有f(x1+x2)=f(x1)f(x2),且f'(0)=1,证明f'(x)=f(x)
设f(x)对任意的实数x1,x2有f(x1+x2)=f(x1)f(x2),且f'(0)=1,证明f'(x)=f(x)
设f(x)对任意的实数x1,x2有f(x1+x2)=f(x1)f(x2),且f'(0)=1,证明f'(x)=f(x)
证明:
(i)设f(x)在定义域内恒不为零,
由原式得:|f(x+y)|=|f(x)|*|f(y)|
从而:ln|f(x+y)|=ln|f(x)|+ln|f(y)|
等式两边同时对y求导得:(x+y)'f'(x+y)/f(x+y)=f'(y)/f(y)+0
移项整理:f'(x+y)=f(x+y)f'(y)/f(y)=f'(y)f(x)
取y=0得:f'(x+0)=f'(x)=f'(0)f(x)=f(x)
(ii)当f(x)在某点处(不妨设在点a处)为零时,即f(a)=0时,
可知对任意的x,成立:f(x+a)=f(x)f(a)=f(x)*0=0
由x的任意性,知x+a能取遍整个实数空间,则f在实数域内恒为0,亦满足f'(x)=f(x)(=0)
综合(i)和(ii)可知,原命题成立
证毕.
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