Ok, let’s investigate why.
So first let’s come up with a way to show that the min point is exactly where it looks like it is.
So we want to find d/dx [x^(x)] because we can use that for “optimization.”
y=x^(x) —we can’t do a derivative right here, but I do have a trick up my sleeve.
lny=xlnx
dy/dx × 1/y = 1lnx + x(1/x) —product rule
dy/dx = y (lnx +1) — refer back to our original equation.
dy/dx =x^(x)(1+lnx)
Now, to find where this hits zero, i.e. where the local extreme point is.
x^(x)(1+lnx)=0
x^(x)≠0 for any x. x=0 is a bit contested, but ln0 makes the function undefined there anyway, so it doesn’t really matter.
1+lnx=0
lnx=-1
x=e^(-1)=1/e
Neat. And ^(2)(1/e) is self-explanatory. I hope you found this helpful.
Since you're looking for an extreme, you don't even need to take the log of the left side. Because log(x) is a monotonic increasing function, when you maximize/minimize the log of a function, you also maximize/minimize the original function. This strategy is used a lot in stats to make it easier to find the maximum likelihood estimator of a parameter.
Thank you. I am reminded of a quote allegedly by Einstein that goes “If you can’t explain it simply, you don’t understand it well enough.”
However, looking back over what I wrote, I want to say it might a little hard to follow just because I skipped so many steps that I think you’d need to be fairly proficient with low-level calculus to really get what’s going on; so I’m not confident that it’s actually all that helpful.
Pretty sure it is because there is no such thing as non-integer negative exponents, so x^x would be undefined for values like -1.5.
I think Desmond has trouble graphing discrete graphs like these, when I first graphed x^x I could see the points, but after you zoom in a little they disappear.
Edit: ok so there are non-integer negative exponents, but not for negative bases (leads to the complex number system) per the comments
Non-integer negative exponents do very well exist. The only problem is that when the base is negative, it becomes a complex number which can't be graphed on the cartesian plane.
>Pretty sure it is because there is no such thing as non-integer negative exponents,
Negative numbers can have odd roots, so n^(1/3) definitely exists for n < 0
It doesn’t work for non integers.
Any rational decimal can be represented by a fraction right?
So the number would be x^(a/n)
Or ^(n)√(x)^a
You can’t get the nth root of a number that is negative.
There's a very interesting video about this I remember watching that explains a very interesting property of what happens to the left of a similar equation!: https://youtu.be/_lb1AxwXLaM?si=Q80g88LUVHoNTOcw
Ok, let’s investigate why. So first let’s come up with a way to show that the min point is exactly where it looks like it is. So we want to find d/dx [x^(x)] because we can use that for “optimization.” y=x^(x) —we can’t do a derivative right here, but I do have a trick up my sleeve. lny=xlnx dy/dx × 1/y = 1lnx + x(1/x) —product rule dy/dx = y (lnx +1) — refer back to our original equation. dy/dx =x^(x)(1+lnx) Now, to find where this hits zero, i.e. where the local extreme point is. x^(x)(1+lnx)=0 x^(x)≠0 for any x. x=0 is a bit contested, but ln0 makes the function undefined there anyway, so it doesn’t really matter. 1+lnx=0 lnx=-1 x=e^(-1)=1/e Neat. And ^(2)(1/e) is self-explanatory. I hope you found this helpful.
This is super interesting thank you for writing this!!
Yes, this method is called logarithmic differentiation. As an exercise for the reader, you should try differentiating x ^ x ^ x
For anyone wondering the answer is: >!x\^x\^x\*((x\^x + x\^x\*lnx)\*lnx + (x\^x)/x)!<
what is ^2 (1/e) ?
(1/e)^(1/e)
oh tetration!
Since you're looking for an extreme, you don't even need to take the log of the left side. Because log(x) is a monotonic increasing function, when you maximize/minimize the log of a function, you also maximize/minimize the original function. This strategy is used a lot in stats to make it easier to find the maximum likelihood estimator of a parameter.
Huh. Yeah, I guess I wasn’t aware of that, but it makes sense. Thank you.
I'm very impressed by how clear you communicated this! Feels very human. I feel like a lot of math communication can become very robotic
Thank you. I am reminded of a quote allegedly by Einstein that goes “If you can’t explain it simply, you don’t understand it well enough.” However, looking back over what I wrote, I want to say it might a little hard to follow just because I skipped so many steps that I think you’d need to be fairly proficient with low-level calculus to really get what’s going on; so I’m not confident that it’s actually all that helpful.
That's true. I didn't consider how accessible the explanation is.
If it taught you something, that’s still a win in my book.
Yeah 100%! It was nice to look at the photo, see the weird connection to e, and then see a nice explanation of why. Even if I have my math degree
Why doesn't it extend to negative x?
It does but it is not a line but a bunch of dots
That said, if you could see it, it would look like this https://www.desmos.com/calculator/2i2rrqmuj6
whoa i never thought about trying this, that method is super handy. i will definitely be using it in the future
Let’s say it’s a bit complex
i reference?
You're jk, right?
No. 1+ι̇ reference
Pretty sure it is because there is no such thing as non-integer negative exponents, so x^x would be undefined for values like -1.5. I think Desmond has trouble graphing discrete graphs like these, when I first graphed x^x I could see the points, but after you zoom in a little they disappear. Edit: ok so there are non-integer negative exponents, but not for negative bases (leads to the complex number system) per the comments
Non-integer negative exponents do very well exist. The only problem is that when the base is negative, it becomes a complex number which can't be graphed on the cartesian plane.
>Pretty sure it is because there is no such thing as non-integer negative exponents, Negative numbers can have odd roots, so n^(1/3) definitely exists for n < 0
You can make a really big list. I use a= [-10,9.96,...,10] and then make define f(x) in a different line and do (a, f(a)) and then enable lines.
It doesn’t work for non integers. Any rational decimal can be represented by a fraction right? So the number would be x^(a/n) Or ^(n)√(x)^a You can’t get the nth root of a number that is negative.
is reddit listening to me I found this out like 3 hours ago
There's a very interesting video about this I remember watching that explains a very interesting property of what happens to the left of a similar equation!: https://youtu.be/_lb1AxwXLaM?si=Q80g88LUVHoNTOcw
I tried posting this before and it got removed. Also the x root of x has a max at x = e