B Something Fun I Stumbled Across

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The discussion highlights the discovery of the intersection point of the function f(x) with the line y=x, which occurs at the golden ratio (φ, φ). It explains the process of finding this intersection through algebraic manipulation and the minimal polynomial associated with φ. Additionally, the limit of f(x) as x approaches infinity is determined to be 1 using L'Hospital's Rule. The derivative of f(x) at the intersection point is also calculated, emphasizing the importance of restricting the domain to positive real numbers. The post concludes with a request for feedback on the mathematical presentation and LaTeX usage.
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Hi! I know all of you might know what I'm about to post, but I just discovered it for myself, and I want to share my enthusiasm.
Let
gif.gif

and
gif.gif
(here, I'll be restricting the domain of f(x) to the positive real numbers.)
Here is a graph of the two, with f(x) in blue and F(x) in black:

upload_2018-5-9_19-1-12.png

1st question: Where does f(x) intersect with the line y=x?

you could write
gif.gif

squaring both sides of the equation, multiplying both sides by x-1, and subtracting x from both sides gives
gif.gif

Factoring x from the LHS and dividing both sides by x leaves you with
gif.gif

This is the minimal polynomial for the golden ratio, or φ, and the minimal polynomial for -φ^-1, or -Φ. This means that the quadratic above has two solutions at φ and -Φ. -Φ cannot be the solution we are looking for, as, as stated above, I am only dealing with f(x) within the domain of the positive real numbers (positive x values only). So, the intersection of f(x) and y=x is at (φ,φ)!

Question 2: What is the value of
gif.gif
?

Using L'Hospital's Rule, we obtain that
gif.gif

calling the limit as x approaches infinity of f(x) "L", then this becomes
gif.gif

and obviously L=1. So
gif.gif
.

Question 3: What is the derivative of f(x) at (φ,φ)?

If we take the derivative of f(x), plug in φ for x, and make sure to remember that φ-1=Φ and that φ^-1=Φ, we simplify:

gif.gif
Question 4 (finale): What is
gif.gif
?
From the fundamental theorem of calculus

gif.gif


So we could rewrite this as:

gif.gif
Hope I made no typos! Sorry if this is too long, but I want to share these interesting facts with y'all. Also, hopefully the type doesn't mess up, I used rendered LaTeX and pasted the images here. Also, the prefix is beginner, as most of the calculus stuff is taught in high school, but I really don't know what this is, so sorry if that is wrong.
 

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Last edited:
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Edits: Stupid typos I made. Fixed.
 
When I "restricted the domain of f to the postitve real numbers" instead of differentiating the f(x) I gave at first, I differentiated the square root of x divided by the square root of x-1. That way the domain of the function and its derivative is restricted to the positive real numbers, and that is why you may have obtained a different answer for the derivative because you used the chain rule on the f(x) I gave, which differentiated f(x) with respect to all values of x.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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