设函数f(x)对任意实数x1,x2,总有f(x1+x2)=f(x1)+f(x2),且f'(0)=1,证明f'(x)=f(x)
问题描述:
设函数f(x)对任意实数x1,x2,总有f(x1+x2)=f(x1)+f(x2),且f'(0)=1,证明f'(x)=f(x)
答
确定题目对?
f(0+0)=f(0)+f(0)
f(0)=2f(0)
f(0)=0≠f'(0)
答
当f'(x)=f(x)时,只有f(x)=e^x
显然满足于:f(x1+x2)=f(x1)+f(x2),是一次函数不满足f'(x)=f(x)条件的
满足于f(x1+x2)=f(x1)*f(x2),f'(0)=1,才有f(x)=e^x的