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

In summary, the problem is to show that if f'(x) = f(x) holds, then f(x) must be some multiple of exp(x). The suggested approach is to show a contradiction if c is not equal to f(0). However, this cannot be done using differential equations, so another method must be used.
  • #1
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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?
 
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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.
 
  • #3


CompuChip said:
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).
 

What is real analysis with exponential functions?

Real analysis with exponential functions is a branch of mathematics that studies the behavior and properties of functions that involve exponential expressions, such as f(x) = ex. It involves the use of tools and techniques from calculus, such as derivatives and integrals, to analyze these functions.

What does it mean for f(x) to equal f'(x)?

When f(x) = f'(x), it means that the function f(x) is equal to its own derivative. This is known as a differential equation and it represents a special type of relationship between a function and its rate of change.

What are some practical applications of real analysis with exponential functions?

Real analysis with exponential functions has many practical applications in fields such as physics, economics, biology, and engineering. For example, it can be used to model population growth, radioactive decay, and interest rates.

How can I solve problems involving real analysis with exponential functions?

Solving problems involving real analysis with exponential functions typically involves using calculus techniques such as differentiation and integration. It is important to understand the properties and behavior of exponential functions and how they relate to their derivatives.

What are the main challenges in studying real analysis with exponential functions?

Some of the main challenges in studying real analysis with exponential functions include understanding the complex behavior of these functions, differentiating between different types of exponential functions, and applying calculus techniques accurately and efficiently.

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