Proving that f(x)=kecx without Assuming f=0

[ptolema]

Homework Statement

Suppose that on some interval the function f satisfies f'=cf for some number c. Show that f(x)=ke^cx for some k without the assumption that f is never 0. Hint: Show that f can't be 0 at the endpoint of an open interval on which it is nowhere hear 0

Homework Equations

f'=cf
(log (abs f))'=f'/f

The Attempt at a Solution

i tried to use the case where f=0, which meant that f'=0. From there, it follows that f(x)=k. That's where i got stuck. I know somehow I have to show that f can be f(x)=ke^cx=0, where k=0, but how? Perhaps I need to figure out how to use the hint. Does the hint have anything to do with Mean Value Theorem?

 

[╔(σ_σ)╝]

Integrate f'/f and then solve for f.