How Does f'(x) Relate to f(x) in Functional Equations?

[chief12]

Homework Statement

function f is differential when x=0,
f'(0) is not equal to zero for all a,b(real Numbers)
f(a+b) = f(a)f(b)

show f'(x) = f'(x)f(x)

Homework Equations

The Attempt at a Solution

f(a+b) = f(a)f(b) for all a,b(real numbers)
f(0), a+b=0
then f(0) = 1

lim x-->0 f(x) = f(0) = 1

then i get stuck

 

[lanedance]

Homework Statement

function f is differential when x=0,
f'(0) is not equal to zero for all a,b(real Numbers)
f(a+b) = f(a)f(b)

show f'(x) = f'(x)f(x)

that would imply f(x) = 1 for all x... which then gives f'(x) = f'(0) = 0
also f'(0) is not equal to zero for all a,b,... does this mean only for a=-b?

are you sure this is how the question was written? try and write thinsg exactly as they are given...

 

[lanedance]

now guessing at what the question actually asked... i would start by considering
f(x+0) = f(x) = f(x)f(0)
this shows f(0) = 1

 

[lanedance]

then consider f(2x)