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Thread: What all can we say about these two functions?

  1. #1
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    What all can we say about these two functions?

    Hello, so I have a problem here - the only information I know about two functions f(x) and g(x) is the following:

    f'(x)*g(x) = f(x)*g'(x) and g(x) cannot equal 0 on the interval (a,b)

    What can we say about these two functions and how they are related?

    Here is my thought process so far:

    I'm thinking this can be split into two cases, where f'(x) = f(x) and g(x) = g'(x) OR where f'(x) = g'(x) and g(x) = f(x)

    Then, from there we can say that in the first case, f(x)=g(x)=f'(x)=g'(x) which equals either 0 or e^x, since this is the only function where the derivative can equal itself

    For the second case, we can say that it will be the same function


    Is my reasoning correct? Am I missing something else about they how they may be related? Thanks
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  2. #2
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    Re: What all can we say about these two functions?

    Quote Originally Posted by pizzapie View Post
    Hello, so I have a problem here - the only information I know about two functions f(x) and g(x) is the following:

    f'(x)*g(x) = f(x)*g'(x) and g(x) cannot equal 0 on the interval (a,b)

    What can we say about these two functions and how they are related?

    Here is my thought process so far:

    I'm thinking this can be split into two cases, where f'(x) = f(x) and g(x) = g'(x) OR where f'(x) = g'(x) and g(x) = f(x)

    Then, from there we can say that in the first case, f(x)=g(x)=f'(x)=g'(x) which equals either 0 or e^x, since this is the only function where the derivative can equal itself

    For the second case, we can say that it will be the same function


    Is my reasoning correct? Am I missing something else about they how they may be related? Thanks
    At a first glance you are making a rather stringent assumption. Let's put it in terms of simple algebra. You seem to be saying: 6 x 8 = 6 x 8 or 6 x 8 = 8 x 6. Okay, no problem. However 6 x 8 is also 2 x 24. Thus f'(x) and g'(x) need not either be f(x) or g(x).

    -Dan
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    Re: What all can we say about these two functions?

    How about
    \frac{f'g - fg'}{g^2}=\left(\frac{f}{g}\right)'=0
    What can you say now?
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  4. #4
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    Re: What all can we say about these two functions?

    Hm ok this makes sense - so is there another way to write this or a specific thing we can derive from knowing this?
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  5. #5
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    Re: What all can we say about these two functions?

    How do you know that first equation? (f'g-fg')/g^2?
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  6. #6
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    Re: What all can we say about these two functions?

    You are given f'g=fg' so the numerator is zero. You are told that g is not zero, so you can divide by it (twice).

    You have obviously studied the quotient rule of differentiation to be set the question.
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  7. #7
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    Re: What all can we say about these two functions?

    Quote Originally Posted by pizzapie View Post
    a specific thing we can derive from knowing this?
    What have you differentiated if the answer is zero? What type of function/term have you differentiated?
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  8. #8
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    Re: What all can we say about these two functions?

    Ahhhh ok yes the quotient rule. So we can basically say that at any point in that interval (a,b) that (f/g)'=0?
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  9. #9
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    Re: What all can we say about these two functions?

    Oh, so you're saying that (f/g) must be a constant?
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  10. #10
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    Re: What all can we say about these two functions?

    Exactly.
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