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Thread: Solution for a self-intersecting graph

  1. #1
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    Solution for a self-intersecting graph

    Hello guys,
    I was doing some work with functions, and I came across equations which have $\displaystyle y $ on both sides, such as

    $\displaystyle y=x^{\frac{y}{x}} $

    If $\displaystyle y=x $, then this equation becomes

    $\displaystyle y=x^{\frac{x}{x}} $

    $\displaystyle y=x $

    So from this I guess it can be concluded that one part of the graph of the equation $\displaystyle y=x^{\frac{y}{x}} $ is going to be the line $\displaystyle y=x $. However, there is another part to the graph since all of the inputs $\displaystyle x $ that are greater than 1 have two solutions for $\displaystyle y $ (except for $\displaystyle e $ apparently, according to the graph). Assuming $\displaystyle x=e $ is the only point that has one solution in the domain $\displaystyle x>1 $ of the graph $\displaystyle y=x^{\frac{y}{x}} $, then that is the point the graph intersects itself in a way.

    My question basically is, how is it possible to solve for the value where such a graph (or graphs of this type in general) has only one solution (in other words how to find the point of its self-intersection), because in my example I assumed this point was $\displaystyle e $ because it really looked like it when I zoomed in. Is there a way to prove it without approximation?

    Best,
    Ivan
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  2. #2
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    Re: Solution for a self-intersecting graph

    the equation can be written as

    $$\frac{\ln y}{y}=\frac{\ln x}{x}$$

    so the question is when does

    $$\frac{\ln x}{x}=c$$

    have exactly one solution?
    Last edited by Idea; Sep 20th 2019 at 11:39 PM.
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  3. #3
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    Re: Solution for a self-intersecting graph

    Yeah, so here below is the graph of the equation $\displaystyle \frac{\mathrm{ln}y}{y}=\frac{\mathrm{ln}x}{x} $. My question is how to basically determine the point where $\displaystyle x $ has only one solution for this graph in the domain $\displaystyle x>1 $ (which looks like it's $\displaystyle e $).
    Attached Thumbnails Attached Thumbnails Solution for a self-intersecting graph-ezgif.com-gif-maker.jpg  
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  4. #4
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    Re: Solution for a self-intersecting graph

    Quote Originally Posted by IvanM View Post
    $\displaystyle y=x^{\frac{y}{x}} $

    If $\displaystyle y=x $, then this equation becomes

    $\displaystyle y=x^{\frac{x}{x}} $

    $\displaystyle y=x $
    IvanM,

    if y = x, then every y is to be replaced with x.

    The equation becomes

    $\displaystyle x \ = \ x^{\frac{x}{x}} \ \implies $

    $\displaystyle x \ = \ x, \ \ x \ne \ 0$
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  5. #5
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    Re: Solution for a self-intersecting graph

    This happens at the point where the function

    $$f(x)=\frac{ln x}{x}$$

    has a maximum i.e. at the point where the derivative is zero and that is at $x=e$
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  6. #6
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    Re: Solution for a self-intersecting graph

    Oh I see now, thank you so much for pointing that out!

    Best,
    Ivan
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