Thread: Solution for a self-intersecting graph

1. 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

2. 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?

3. 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 $\displaystyle e$).

4. Re: Solution for a self-intersecting graph

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

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

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

$\displaystyle x \ = \ x, \ \ x \ne \ 0$

5. 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 $x=e$

6. Re: Solution for a self-intersecting graph

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

Best,
Ivan