Real analysis with exponential functions; given f(x) = f'(x)

[rapple]

Homework Statement

f(x)=f'(x) for all x in R
S.T there exists a c in R such that f(x) = c exp(x) for all x

Homework Equations

The Attempt at a Solution

By defining g = f/c, I was able to show that c= f(0)
But i am also supposed to show that c Not equal to any other value

I tried assuming that c=f(1) and cannot find a contradiction unless I plug it into f(1)=f(1).exp(1).

I am supposed to use Mean value Thm to show the contradiction. How do I go about doing it?

 

[CompuChip]

I don't see what your point is. I assume that "S.T" means "show that" rather than the usual "such that".
You are not supposed to show that c is unique. I think you are supposed to show that if f'(x) = f(x) holds, then f(x) must be some multiple of exp(x).

How you are going to do this depends on what you know and can use. For example, using the theory of differential equations you can easily show that the equation f'(x) = f(x) with f(0) = c has a unique solution.

 

[rapple]

I don't see what your point is. I assume that "S.T" means "show that" rather than the usual "such that".
You are not supposed to show that c is unique. I think you are supposed to show that if f'(x) = f(x) holds, then f(x) must be some multiple of exp(x).

How you are going to do this depends on what you know and can use. For example, using the theory of differential equations you can easily show that the equation f'(x) = f(x) with f(0) = c has a unique solution.

This is in real analysis in the section on exponential functions. So I can't use differential equations.

The only way, I suppose, is to arrive at a contradiction if c not= f(0).