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

  • Thread starter ptolema
  • Start date
In summary, to show that f(x)=kecx for some k without the assumption that f is never 0 on an interval where f'=cf, we must use the hint given and show that f cannot be 0 at the endpoint of an open interval where it is nowhere near 0. This can be done by integrating f'/f and solving for f, which will result in f(x)=kecx. The hint may involve using the Mean Value Theorem.
  • #1
ptolema
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Homework Statement



Suppose that on some interval the function f satisfies f'=cf for some number c. Show that f(x)=kecx 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)=kecx=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?
 
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  • #2
Integrate f'/f and then solve for f.
 

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

What is the formula for f(x)?

The formula for f(x) is f(x) = ke^(cx), where k and c are constants.

How do you prove that f(x) = ke^(cx)?

To prove that f(x) = ke^(cx), you must show that the function satisfies the definition of an exponential function, which is f(x) = ab^x, where a and b are constants. By substituting ke^(cx) for f(x), you can show that it satisfies the definition.

Can you prove f(x) = ke^(cx) without assuming f(x) = 0?

Yes, it is possible to prove f(x) = ke^(cx) without assuming f(x) = 0. This can be done by using the properties of exponential functions and mathematical induction. By showing that the function satisfies the exponential function definition for any value of x, you can prove that f(x) = ke^(cx) without any assumptions.

What is the significance of proving f(x) = ke^(cx)?

Proving f(x) = ke^(cx) is significant because it shows that the function is an exponential function. This allows for further analysis and understanding of the behavior of the function. It also allows for the use of exponential function properties and equations to manipulate and solve for the function.

Are there any real-life applications of proving f(x) = ke^(cx)?

Yes, there are many real-life applications of proving f(x) = ke^(cx). For example, in biology, exponential growth and decay can be modeled using this function. In economics, the compound interest formula can be derived from this function. It is also used in physics and engineering to model processes such as radioactive decay and population growth.

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