Something Fun I Stumbled Across

[Thinkaholic]

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

and

(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:

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

you could write

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

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

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

?

Using L'Hospital's Rule, we obtain that

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

and obviously L=1. So

.

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:

Question 4 (finale): What is

?
From the fundamental theorem of calculus

So we could rewrite this as:

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.

 

[Thinkaholic]

Edits: Stupid typos I made. Fixed.

 

[Thinkaholic]

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.

 

[DrClaude]

Also, hopefully the type doesn't mess up, I used rendered LaTeX and pasted the images here.

If you know LaTeX, then you should post LaTeX directly: https://www.physicsforums.com/help/latexhelp/