f(x)在x=0处可导且f'(0)=ln2,且对任意的x,y∈R有f(x+y)=f(x)f(y),求f(x)

Question

f(x)在x=0处可导且f'(0)=ln2,且对任意的x,y∈R有f(x+y)=f(x)f(y),求f(x)

Answer 1

f(0+0)=f(0)f(0),f(0)=1或f(0)=0
假如f(0)=0,{f(0+h)-f(0)}/h={f(0)f(h)-f(0)}/h=0,则有f′(0)=0≠ln2,所以f(0)≠0
f(x+h)-f(x)=f(X)f(h)-f(x)=f(x)(f(h)-1)=f(x)(f(h)-f(0))
{f(x+h)-f(x)}/h=f(x)(f(h)-f(0))/h,h→0,有f′(x)=f(x)ln2, dy/dx=yln2,x=0,y=1
dy/y=ln2dx,积分得lny=xln2+c, x=0,y=1,c=0
lny=xln2,
y=2^x