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Math Help - Suppose this, show that:

  1. #1
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    Suppose this, show that:

    PROBLEM:

    Suppose that a function f satisfies the following conditions for all real values x and y:

    i. f(x+y)=f(x)*f(y)
    ii. f(x)=1+xg(x), where \lim_{x\to\0}g(x)=1

    Show that the derivative f'(x) exists at every value of x and that f'(x)=f(x).

    ATTEMPT:

    I honestly don't have much of a clue here. Instead, I've tried to ask myself questions to get some idea going but to no avail. First, I tried to make sense of what the conditions imply and discovered that because x and y belong in the same set of real numbers without restriction, then everything said about x is also true about y.

    Therefore, f(y)=1+yg(y), where \lim_{y\to\0}g(y)=1

    I thought, maybe, if I could rewrite condition (i) with a substitution in terms of (ii), I might get somewhere.

    Any thoughts?
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  2. #2
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    Re: Suppose this, show that:

    Hey Lambin.

    Hint: Consider the exponential function: Can you use it to show the first property (as a start)?
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  3. #3
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    Re: Suppose this, show that:

    Lambin

    Did you realize the important properties of your function?

    1. f(0) = 1 : Indeed if you put x = y =0 then you will get it.

    2. f(x) is different than zero for all real numbers x… Indeed if for a real number y we have
    f(y) =0 then f(x+y) = 0 for all values of x .i.e for all real numbers ..that is absurd…

    3. f(x-y) = f(x) /f(y) indeed substitute x = x-y and you get f(x)=f(x-y)f(y) then…

    4. f(x) >0 for every real number x . indeed substitute x = y and you will get it….

    The only known function from functional Analysis to have all these properties is the function f(x) = e^x which is differentiable for all real numbers x and f’(x) = f(x).

    MINOAS
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  4. #4
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    Re: Suppose this, show that:

    I think that Lambin was looking for a more direct proof.

    If you look at the definition f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} and work with the given properties of f, I think you'll get to the result pretty quickly: start by substituting f(x)f(h) for f(x+h).

    - Hollywood
    Thanks from Lambin
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  5. #5
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    Re: Suppose this, show that:

    Hollywood posted his reply while I was typing, but here it is anyway:

    Suppose this, show that:-mhfcalc3.png
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  6. #6
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    Re: Suppose this, show that:

    Quote Originally Posted by johng View Post
    Hollywood posted his reply while I was typing, but here it is anyway:

    Click image for larger version. 

Name:	MHFcalc3.png 
Views:	7 
Size:	11.8 KB 
ID:	27725
    Oh, this is fantastic! You know, I actually did peer into the definition of the derivative but must have overlooked its resemblance to condition (i), otherwise I would have been able to figure it out. Intuitively, I thought it might be possible that f'(x) = f(x) because the first condition was simulating an exponential function.

    Thanks again for all these replies.
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