"I really love this question, let's... I really love this question, let's have a look. I really love this question, let's have a look." - blackpenredpen
While most of this went completely over my head, it was a fun watch, and I am quite proud of myself that my first thoughts were "Are they positive numbers, and are they whole numbers?"
The 2 most important parts of the video is recognizing that you have to take the derivative to find the min and max and then realizing that u can use the Lambert W function, everything else is following up on those 2 things
found this channel last night. i used to "teach"/tutor math in college and have always been passionate about teaching despite not being very good at it, so I love watching you explain intermediate math in very straightforward ways. but, i personally tapped out of math education in late calculus and linear algebra, and to see a video on more advanced subjects that I'M not even familiar with like the Lambert W... it made me feel very warm and cozy. This is a wonderful channel and I'm delighted to have found it.
@@FourOneNineOneFourOne I did learn some of it in high school and even at university, but I graduated from university 5 years ago, my maths class had finished 2-3 years prior, and I have been working in a completely unrelated area. And in my life the only areas of maths I use are arithmetics and trigonometry 😂 so, the other things get forgotten easily. I think we didn't reach this difficulty of calculus at high school, or, if we did, it's gone from my memory, because I understood about half of the video at most. I don't remember any of the functions, sadly.
@@FourOneNineOneFourOne At least in my case, Calc wasn't even offered unless you qualified to do College Courses while still in Highschool. I got through Calc 1 just fine, but most of what he was showing here is not stuff we really covered. This is the first time I ever heard of W(xe^x)=x, and I would have literally no idea how to parse something like that in a casual way. While I have worked with Log/ln/e a bit back then, it's been over a decade, and I was never fond of dealing with those at all, so I just lost it completely.
@@FourOneNineOneFourOne What if one has no aptitude for math? As a high school student, or today more than 30 years later, I could no more understand what he is explaining then I could run 400m in 48 seconds (I never broke the 50s barrier 50.7, and only passed algebra as a senior.) The ability to do high school math, beyond basic algebra, is a mental gift, and is no different than the ability to any number of other things. And no amount of hard work in the world will overcome a lack of ability. If you can do this stuff, congrats to you, but no one “should” be able to do anything you can do.
@@JonesyTerp1 "no amount of hard work in the world will overcome a lack of ability". I'm not sure I agree, particularly for all the activities you mentioned, but, putting that aside... The problem is, even if you have the "innate ability", for most (many?) things, you still have to do the "hard work" before you are proficient enough to be convinced you have the ability. So, to find out if you have the ability to do "a thing", you have to take a leap of faith and put in the hard work to find out. One would hope you enjoy "the thing" so if it turns out you can't do it, at least you had fun trying. It comes down to the point, does the hard work "develop" the aptitude, or does it "expose" it? I think there are some people in each group, along with a group who work hard and still can't, and another group who never seriously try.
for a lot of people the concept of powers might not be that clear or straightforward. Younger students who may have an interest comes to mind. Of course however, most high-schoolers or college level students (who take math courses), will see it very clearly.@@xlr8_bs514
Using the Lagrange multiplier will give a system of equations identical to solving using the methods of single-variable calculus, ie the solution x+xln(x)=8.
@@TomJones-tx7pb but what I stopped with is an implicit function. Explicit solution presents the way to directly calculate the value for a given argument rather than to bruteforce it
Same for me. I also solved the problem before watching the video and got stuck with x ln(x) + x = 8 but I just type it into Wolframalpha and see that it has no "nice solution". In my opinion, there is no benefit in expressing this equation with the W function. The equation x ln(x) + x = 8 can already easily be solved by a computer and for a human the W function does not give you any additional information
@@adammizaushev Good point for classical math, if you have a pre-calculated W function, but modern way of solving is using a computer to get as accurate an answer as you desire with numerical analysis.
@@TomJones-tx7pb Yeah, I was only speaking in the matter of getting intuitive, natural understanding of a function to a human, which is got by its explicit expression, rather than implicit one
(even -ωₙ)^(even ωₙ+8) where n is arbitrary Because a negative when exponentiated to a positive even is positive, this produces a value greater than ωₙ₂ Where n⇐∞, the maximum approaches Ω.
I mean, if you had it so X could be negative, the max is theoretically infinity, but that min is theoretically negative infinity, which makes the question irrelevant. Knowing that is actually why I clicked on the video in the first place.
@@TheSkullkid16 Exactly. You end up with an equation of -(n-8)+n=8 and then have (-(n-8))^n. The limit would basically be negative infinity to the power of infinity plus 8.
honestly, I understood everything you said (despite this being the first time I've heard about lambert w function) and this made me so happy (also, the fish you draw are amazing)
@@abrahamholleran4162 so like, you get that my skepticism came exclusively from the idea that e^{W(100e)-1} is a natural number right? Like, sqrt(2) is "close" to 1.4, but the fact that it's irrational is kind of a really important thing about it
@@frimi8593 They weren't replying to you, but rather to the original comment. I'm sure they would assume that you were doubting it being an integer. (It can be difficult to tell what people mean, since some use the word "equals" too loosely. You could say something like "exactly 24" or just explicitly doubt it being an integer.)
Exceptional explanation of the unique "W" Function! However, I still need to exponentially repeat your Steps to the Solution (or repeat the Steps to the maximum number allowed) --- in keeping with the ultimate objective of this proposal (computing the maximum)! Alas, I am exhausted from this exhilarating mathematical exercise. Bravo! Thank you, sir blackpenredpen!
Yes, x=0 is a local minimum because to the left the value of P is unbounded. That means that I think the result we found is not a global max but a local max. Wolfram is mistaken 😮 because it calls it a global max. Or have we told Wolfram that x>=0 ?😊
Wolfram alpha probably ignores negatives entirely here because any noninteger negative x can't be raised to the power 8-x when restricted to the reals.
As soon as I saw the question I noticed there's no upper limit because we can have absurdly large negative numbers for X (or Y). Once he put the constraint in that they had to be positive values, I was sure the answer was x=y=4. I just assumed the answer was going to be integers. Don't know why I fell into that trap, it just felt intuitive.
@@KenFullman Exactly, I was thinking the same thing. Because, you can have the equation as such: y=n, and x= -(n-8) For example, you could have y=1,000,008. x would then equal -1,000,000 and the answer would still equal 8, and you'd be left with -(1,000,000)^1,000,008 which would be absolutely huge, and can still get bigger, into infinity. Because you'd have -(n-8)+n=8 and then (-(n-8))^n. As long as you can have y be 8 more than a negative value of x, you can get infinitely higher values for x and y.
@@KenFullman If the equation had been x*y instead of x^y, then that intuition would be correct, and I frequently encounter situations where it's useful to optimize x*y by setting x=y (maybe you do too). So maybe that explains how you fell into that trap. :)
It's always about local extremum, cause blackpenredpen used Fermat's theorem that states that *local extrema* in the interior of a domain must occur at points where the derivative equals zero or undefined.
You have a lovely collection of white board pens, but at 7:00 when you write over the black X with the red pen it made me feel a little bit ill inside. You then cured this illness with a lovely explanation of a powerful tool I'm sure I'll use again at some point.
Bprp, I have one more question. You know how we solved this problem of maximizing x^y w/ x+y=8 using derivatives? Suppose we restricted the domain of x,y to be natural numbers again. Could we then use discrete calculus to solve this problem. That is forward difference/backward differences are now analogies to the first derivative?
I went through my YT feed and marked all the politics, sports, pop culture, and crime stories as "not interested", and now finally I'm seeing interesting content like this in my feed. What took me so long?😀
@@richatlarge462I get not being interested in pop culture, sports and stuff, but why did you get rid of politics? Politics are highly important. Its what governs us and bad politics will have bad consequences in society and in people's lifes. When people aren't interested in politics, you eventually get leadership like Russia. So please take an interest in it. I'm sure you make use of your right to vote.
Hmm this video demonstrates the difference in thinking between an engineer and a mathematician. I paused the video when the question was posed, opened excel, used one cell as an input for X, defined another cell as Y by making it 8 - the X cell, then defined a 3rd cell as the X cell to the power of the Y cell. At this point I used the solver function in excel to maximise the value of the 3rd cell by altering the value of the X input cell. Got the answer in less than a minute by brute forcing it 😅. I will admit blackpenredpen's method was more eloquent.
I'm right there with you. I might have done a column for x, another for 8-x and one more for the function. Range 3-5 with 0.05 increments, maybe refine around the maximum and called it good at 3.51.
Tried solving using desmos. Plotted x+y=8 first. Then plotted x^y = some constant b. The value of b where the two graphs only have one point of intersection is the maximum value of the function, and that came out to be approximately 280.902799
hey @blackpenredpen, can you make a shirt that shows a white and green pencil and says "I really hate this question, let's not take a look ~whitepencilgreenpencil"? pweeeasee???
f(x) = x^(8-x) ln(f(x))' = f'(x)/f(x) the max is at f'(x) = 0, and f(x) can't be infinity so the max is at ln(f(x))' = - ln(x) + (8 - x)/x = 0 Ask a computer for an approximation
I feel like we've been duped, he uses a black pen, red pen AND A BLUE PEN!!! Seriously though the explanation made a lot of sense and the way you used three pens shows you're a pro at this lol.
This can also be solved using binary search import math low = 0 high = 8 delta = 1e-6 while abs(high-low) > delta: mid = (low+high)/2 first = math.pow(mid-delta, 8-mid+delta) second = math.pow(mid, 8-mid) third = math.pow(mid+delta, 8-mid-delta) if firstthird: high = mid else: break print(math.pow(low, 8-low))
6:10 "Fishes" is correct here. It's a weird corner of English: it's "fishes" if they're easily countable, "fish" otherwise. No stranger than infinities, I guess.
Fishes is 'multiple fish of different species', if there are multiple fish of the same species, then they're fish. 6 fish, all salmon, vs 3 fishes, salmon, trout, mackerel.
@@Azmodon If you order fish for dinner, you might get 2 fishes on your plate. It's not just types of fish, it's simply countable vs uncountable. At least, per my OED Style Guide.
It's one of those funky words that is really dependent on the article used in front of it like the fishes vs fish like you were saying on it denoting a type but nothing more it seems
As others have pointed out, you could use the same method to get x=k/W(k*e) for the max of x^y when x+y=k. The case of k=8 is interesting though, because 4^4 and 3^5 are so close together. We were already pretty close with 3.5^4.5.
My youngest son hit his high school math teacher with a similar answer as there was no criteria on a range for the problem, as with her and must all be positive.
Can you do an all-in-one calc question again? But please do a d/dx this time and put integrals inside Edit: and i know he did it 4wk ago but i want one with a derivative
Quick education on fish or fishes: a fish is one singular fish, the fish are multiple fish from the same species of fish, the fishes are multiple fish from multiple species of fish :)
I got pretty far into the question before I eventually got stuck... Of course the Lambert W function was involved lmao, I still dont know how to properly use that thing
I like how he shows that ChatGPT is unable to find the solution at the end. Sure, it can spit out an explanation for how to find the solution, because those sorts of explanations are in its training data, but it can't process the solutions for itself because it doesn't "understand" what the explanation means. It can't create a mathematical model, much less perform the operations to process it, because it has no understanding or procedure for that. It only recognizes that this sort of problem resembles the maxima/minima problems from its training data, and regurgitates a procedure. It's useful to recognize the limitations of the language model.
I don't pay for v4 but v3.5 has always been awful at calculus and has only gotten worse as they have limited the product. I love learning how to solve problems (hence why I am here) but I have yet to find a good AI solution.
Here wrote a little python script don’t know if this will help max_num = 0; x = 1 y = 7 while (int(x+y) == 8): print(x+y) if (max_num < x**y): max_num = x**y; x+=0.0001 y-=0.0001 print(x+y) print(max_num)
It may be because I never studied it so I don't know much about it, but I don't really like the Lambert function. Since there's no way to determine its value without using wolpharm alpha or a software, it's not really explicitating or finding the value of X. I feel it's just writing an implicit equation in a different way that's still an implicit equation. It's like saying "there's no solution but here's one that isn't really one".
That's an inherent problem with inverses. Think of it like roots and logs where there's no other way to express those numbers. For example, √2 and ln(2) have no nice ways of expressing them, and you generally need a calculator to compute their values. Not much you can do about it when you have an irrational answer.
@@sethb124 but at least with square roots you can have a general idea of their value, you can at least figure out a min/max borders, like, you know that square root of 2 is somewhere between 1 and 2, you can, like, try out 1,1 * 1,1 then 1,2 * 1,2 and so on and figure out a few decimals this way. None of that with the Lambert function. Either you leave it at that or use a software. At least as far as I know, there's no way to even figure out its value, better off go back to the implicit expression and try out values there.
@AcaciaAvenue Yeah, that's true. It's not super easy to estimate, but for problems like these, it's the most "exact" answer you'll get. It's not very intuitive, but I'd argue neither are lnx and e^x. It's still a valid answer because there are ways of estimating it (or else computers couldn't do it), it's just not an intuitive function.
@@sethb124That's the point I wanted to make. It's not an "exact" answer to me if I have no means of giving even an estimate numerical value. It's just picking something with no answer and writing it differently, but still with no answer. If there were some means to calculate an estimate value of the lambert function without feeding it to a software and tell it "do it for me", then I will change my mind but I looked it up but there seems to be none. Therefore I don't see the utility of the Lambert function.
Totally agree. Just seems like a fix to me. And calling it exact is a stretch too. I solved this by implicit differentiation and Newton-Raphson. Didn't take long at all.
Much the same - I don't know if it is easier - but from the differential... -ln x -1 +8/x = 0 ln x + 1 = 8/x e.x = e^(8/x) e.8 = (8/x)e^(8/x) W(e.8)=2.26282674=8/x x=3.535401 x^(8-x)=280.9045567
duh: without excluding negative numbers: (-infinity)^(infinity+8) because both are even, it gives a positive answer which is infinitely large, thus providing a max that has no end
i used desmos for non whole numbers, and i used this equation: y = x^(8-x) I got 3.535^(8 - 3.535), which is approximately 280, which is greater than 4^4, or 256.
chat gpt just take info from the internet/database and provide an interpretation. if the database doesnt have the answer, chatgpt wont have it. it doesn't "think" on its own. edit: actually it DOES learn, for it's an AI, but it specializes in predicting what the response to your question is base on the database of information it's accessing. But i dont think it can do math.
@@kingpetunfortunately chatgpt doesn't have the capabilities to do calculus but it might be able to do basic arithmetic, they could integrate something like Wolfram alpha into chatgpt one day
@@maddenbanh8033chatgpt has emergent problem solving as a result of understanding human language. Gpt-4 is able to use that problem solving to load arithmetic into premade calculators, like programming languages or Wolfram, in order to “solve” calculus and other math problems. It figures out what steps are necessary, and loads those steps into something that can actually get a confident result. Then it takes that result and moves on to the next step
I'm glad that I was able to solve it by myself. I found the maximum value of x by using newton's method directly as opposed to the lambert w function though (I vaguely recall that you find the values of W using newton's method anyway right?) .
This really reminded me how I hate this weird way to write 1, so confusing. And using chained fractions is ugly and unnecessarily confusing. And writing e^1 instead of e is curious.
Another Chatgpt response when I asked it to solve using calculus: Certainly! To solve this problem using calculus, we first express \(y\) in terms of \(x\) using the constraint \(x + y = 8\), which gives us \(y = 8 - x\). Our function to maximize is therefore \(f(x) = x^{8 - x}\). To find the maximum value, we need to: 1. Take the derivative of \(f(x)\) with respect to \(x\). 2. Set this derivative equal to zero and solve for \(x\) to find critical points. 3. Check these critical points to determine which one gives the maximum value of \(f(x)\). The derivative of \(f(x)\) can be challenging because it involves both the power rule and the chain rule. Let's calculate it. from sympy import symbols, diff, solve # Define the symbol x x = symbols('x') # Define the function f(x) = x^(8 - x) f = x ** (8 - x) # Take the derivative of f with respect to x f_prime = diff(f, x) # Find the critical points by solving f'(x) = 0 critical_points = solve(f_prime, x) # Show the derivative and critical points f_prime, critical_points The critical point for maximizing the function \(f(x) = x^{8 - x}\) is \(x = \exp(-1 + \text{LambertW}(8e))\), and the maximum value of \(f(x)\) at this point is approximately 280.905. This result confirms our earlier graphical analysis, demonstrating that calculus provides a precise way to determine the maximum value of \(x^y\) given the constraints \(x, y \geq 0\) and \(x + y = 8\).
Thank you,! I wasted some time trying to solve this only with derivates and found only (0, 8) and (8,0) 😞. I don't even remember the Lambert W from school. Very nice!
Well, do we assume x and y positive? x=-50 y=58 still produces an x+y=8, meaning any arbitrarily big and even negative x will solve to a greater arbitrarily big value of x^y.
or if you're me and don't know calc but you know python and are a fan of bruteforcing, you write this lol def max(iteration_amt, sum_amt): i, x, y, max_found = 0, 0, 0, 0 while i < sum_amt: x = sum_amt - i y = sum_amt - x if x**y > max_found: max_found = x**y best_nums = [x, y] i += iteration_amt return max_found, best_nums print(max(0.001, 8)) (decrease iteration amount for higher decimal precision)
![I.N "HALLUCINATION" | [Stray Kids : SKZ-PLAYER]](http://i.ytimg.com/vi/n5B5q1Hwt_U/mqdefault.jpg)
If x^y=8 and x is bigger than 1, find the minimum of x+y (Lambert W function)
ruclips.net/video/K6kL3sKw36k/видео.html
I literally thought I was glitching with how many times I heard him say 'I really like this question" lmao
Yeah, I really like this question
he really likes the question
7:56 you are
I thought he was glitching.
i wonder if he likes the question
"I really love this question, let's...
I really love this question, let's have a look.
I really love this question, let's have a look."
- blackpenredpen
It’s a dude that holds a little lapel mic. I don’t think he’s too concerned with edit quality.
I don't know about editing, but I think he really loves this question
where are his other videos with that start lol
I love this question too!
He really loves this question
lambert W function be like :
input→fishy fish
output→fish
😂😂😂
I see what you did there
More like fish*e^fish 😂
BPRP’s ‘lil fish drawings are so cute. Fish*e^fish return fish for life.
lambert w function: defishifier
While most of this went completely over my head, it was a fun watch, and I am quite proud of myself that my first thoughts were "Are they positive numbers, and are they whole numbers?"
The 2 most important parts of the video is recognizing that you have to take the derivative to find the min and max and then realizing that u can use the Lambert W function, everything else is following up on those 2 things
@@sfglim5341 I understood it all till we hit that lambert W thing. I have never seen it before
@@jack002tuberits a very handy tool in college/uni algebra and calculus.
@@jack002tuber It's just a function, you don't need to delve into it too much
found this channel last night. i used to "teach"/tutor math in college and have always been passionate about teaching despite not being very good at it, so I love watching you explain intermediate math in very straightforward ways. but, i personally tapped out of math education in late calculus and linear algebra, and to see a video on more advanced subjects that I'M not even familiar with like the Lambert W... it made me feel very warm and cozy. This is a wonderful channel and I'm delighted to have found it.
This channel has helped me see the usefulness of the product log in some obscure situations, but it comes up frequently enough to remember it.
As a non-mathematician I understood nothing but the way you explain is so cool that it made me more enthusiastic for maths😊
as non-mathematician you should still know highschool maths, unless you're still in school, then I guess you'll still learn about this.
@@FourOneNineOneFourOne I did learn some of it in high school and even at university, but I graduated from university 5 years ago, my maths class had finished 2-3 years prior, and I have been working in a completely unrelated area. And in my life the only areas of maths I use are arithmetics and trigonometry 😂 so, the other things get forgotten easily. I think we didn't reach this difficulty of calculus at high school, or, if we did, it's gone from my memory, because I understood about half of the video at most. I don't remember any of the functions, sadly.
@@FourOneNineOneFourOne At least in my case, Calc wasn't even offered unless you qualified to do College Courses while still in Highschool. I got through Calc 1 just fine, but most of what he was showing here is not stuff we really covered. This is the first time I ever heard of W(xe^x)=x, and I would have literally no idea how to parse something like that in a casual way.
While I have worked with Log/ln/e a bit back then, it's been over a decade, and I was never fond of dealing with those at all, so I just lost it completely.
@@FourOneNineOneFourOne What if one has no aptitude for math? As a high school student, or today more than 30 years later, I could no more understand what he is explaining then I could run 400m in 48 seconds (I never broke the 50s barrier 50.7, and only passed algebra as a senior.) The ability to do high school math, beyond basic algebra, is a mental gift, and is no different than the ability to any number of other things. And no amount of hard work in the world will overcome a lack of ability. If you can do this stuff, congrats to you, but no one “should” be able to do anything you can do.
@@JonesyTerp1 "no amount of hard work in the world will overcome a lack of ability". I'm not sure I agree, particularly for all the activities you mentioned, but, putting that aside...
The problem is, even if you have the "innate ability", for most (many?) things, you still have to do the "hard work" before you are proficient enough to be convinced you have the ability.
So, to find out if you have the ability to do "a thing", you have to take a leap of faith and put in the hard work to find out. One would hope you enjoy "the thing" so if it turns out you can't do it, at least you had fun trying.
It comes down to the point, does the hard work "develop" the aptitude, or does it "expose" it? I think there are some people in each group, along with a group who work hard and still can't, and another group who never seriously try.
At the very start for (-4)^12 it is the same as (-2)^24, for anyone wondering just how much bigger it is.
Isn't that common sense lol
Or more directly, (1024)^2.4.
Which is slightly more than 16 million, because 2⁴ = 16, and 2²⁰ is 1024² > 1,000²
So yeah, I'm not arguing with "bigger" 😉
for a lot of people the concept of powers might not be that clear or straightforward. Younger students who may have an interest comes to mind. Of course however, most high-schoolers or college level students (who take math courses), will see it very clearly.@@xlr8_bs514
@@xlr8_bs514ikr
Using Lagrange multipliers should solve the problem.
x=1/lambert_w(1) and y=8-1/lambert_w(1)
for the ignorant, forgetful, and lazy like myself:
en.wikipedia.org/wiki/Lagrange_multiplier
Using the Lagrange multiplier will give a system of equations identical to solving using the methods of single-variable calculus, ie the solution x+xln(x)=8.
I thought of this method too!
Considering x >= 0, y >= 0 on my own, I got stuck with x(ln(x) + 1) = 8
Thank you for introducing the W function
yeah but you already had a function, so rephrasing it as another function is not really solving anything.
@@TomJones-tx7pb but what I stopped with is an implicit function. Explicit solution presents the way to directly calculate the value for a given argument rather than to bruteforce it
Same for me. I also solved the problem before watching the video and got stuck with x ln(x) + x = 8 but I just type it into Wolframalpha and see that it has no "nice solution". In my opinion, there is no benefit in expressing this equation with the W function. The equation x ln(x) + x = 8 can already easily be solved by a computer and for a human the W function does not give you any additional information
@@adammizaushev Good point for classical math, if you have a pre-calculated W function, but modern way of solving is using a computer to get as accurate an answer as you desire with numerical analysis.
@@TomJones-tx7pb
Yeah, I was only speaking in the matter of getting intuitive, natural understanding of a function to a human, which is got by its explicit expression, rather than implicit one
I did not notice that x and y should be nonnegative, else you could have (-2k)^{8+2k} which goes to infinity.
(even -ωₙ)^(even ωₙ+8)
where n is arbitrary
Because a negative when exponentiated to a positive even is positive, this produces a value greater than ωₙ₂
Where n⇐∞, the maximum approaches Ω.
@@Enlightenment0172!??
I mean, if you had it so X could be negative, the max is theoretically infinity, but that min is theoretically negative infinity, which makes the question irrelevant. Knowing that is actually why I clicked on the video in the first place.
I was thinking just that!
@@TheSkullkid16 Exactly. You end up with an equation of -(n-8)+n=8 and then have (-(n-8))^n. The limit would basically be negative infinity to the power of infinity plus 8.
See, the mistake at the end is that you asked ChatGPT 3.5, you should have asked ChatGPT 8/W(8e).
Elite comment. Simply beautiful
Underrated comment
W COMMENT THIS DESERVES TO BE POPULAR
THE creativity omg
🎉
This is so underrated
I often find your videos a bit beyond my skill level but I found this one really approachable and fun. thanks for this!
yeah this guy is using calc 1 and maybe calc 2
honestly, I understood everything you said (despite this being the first time I've heard about lambert w function) and this made me so happy
(also, the fish you draw are amazing)
The most fascinating about this, is if you do this with X + Y = 100, the maximum will be 24^76, which is totally asymmetrical
You mean to tell me that e^{W(100e)-1} is equal to 24?
Close - the x is still e^(productlog(100e)-1) or 23.947
@@abrahamholleran4162 so like, you get that my skepticism came exclusively from the idea that e^{W(100e)-1} is a natural number right? Like, sqrt(2) is "close" to 1.4, but the fact that it's irrational is kind of a really important thing about it
W(e)=1 so it's a shame there aren't identities for cases a constant precedes e.
@@frimi8593 They weren't replying to you, but rather to the original comment. I'm sure they would assume that you were doubting it being an integer.
(It can be difficult to tell what people mean, since some use the word "equals" too loosely. You could say something like "exactly 24" or just explicitly doubt it being an integer.)
This is a great video and you are a wonderful mathematical communicator! Thank you for creating this!
Exceptional explanation of the unique "W" Function! However, I still need to exponentially repeat your Steps to the Solution (or repeat the Steps to the maximum number allowed) --- in keeping with the ultimate objective of this proposal (computing the maximum)! Alas, I am exhausted from this exhilarating mathematical exercise. Bravo! Thank you, sir blackpenredpen!
didnt understand a word of this but i am now holding a W and a bunch of fish
Always fun mentally juggling x^y | x+y = c among the integers; enjoyed seeing the solution for the reals with a refresher on the W function
pretty straight forward
I'm happy to say that this is the first problem from your videos that I have been able to solve !🎉
This is super fun to watch. I had no idea about the W function.
Great explanation. But what's up with your fish having eyebrows?
Fishes also deserve to have personality 😢
these are the most mischievous fish i've ever seen
10:03 Keep in mind rationalizing the denominator. That case is not needed to put W() in the denominator.
I'm a computer scientist who hasn't done math in a couple of years, it felt good to get this on my own 😌
Yes, x=0 is a local minimum because to the left the value of P is unbounded.
That means that I think the result we found is not a global max but a local max. Wolfram is mistaken 😮 because it calls it a global max.
Or have we told Wolfram that x>=0 ?😊
Wolfram alpha probably ignores negatives entirely here because any noninteger negative x can't be raised to the power 8-x when restricted to the reals.
As soon as I saw the question I noticed there's no upper limit because we can have absurdly large negative numbers for X (or Y). Once he put the constraint in that they had to be positive values, I was sure the answer was x=y=4. I just assumed the answer was going to be integers. Don't know why I fell into that trap, it just felt intuitive.
@@KenFullman Exactly, I was thinking the same thing. Because, you can have the equation as such: y=n, and x= -(n-8) For example, you could have y=1,000,008. x would then equal -1,000,000 and the answer would still equal 8, and you'd be left with -(1,000,000)^1,000,008 which would be absolutely huge, and can still get bigger, into infinity. Because you'd have -(n-8)+n=8 and then (-(n-8))^n. As long as you can have y be 8 more than a negative value of x, you can get infinitely higher values for x and y.
@@KenFullman If the equation had been x*y instead of x^y, then that intuition would be correct, and I frequently encounter situations where it's useful to optimize x*y by setting x=y (maybe you do too). So maybe that explains how you fell into that trap. :)
It's always about local extremum, cause blackpenredpen used Fermat's theorem that states that *local extrema* in the interior of a domain must occur at points where the derivative equals zero or undefined.
You have a lovely collection of white board pens, but at 7:00 when you write over the black X with the red pen it made me feel a little bit ill inside. You then cured this illness with a lovely explanation of a powerful tool I'm sure I'll use again at some point.
all solutions to 2^x-3x-1=0 (transcendental equation)
ruclips.net/video/GJbzsmccFtw/видео.html
Bprp, I have one more question. You know how we solved this problem of maximizing x^y w/ x+y=8 using derivatives? Suppose we restricted the domain of x,y to be natural numbers again. Could we then use discrete calculus to solve this problem. That is forward difference/backward differences are now analogies to the first derivative?
🎉🎉🎉
The finest value of X I’ve found is 3.53540103 but I have no clue how to express the limit of x.
I think the essence of this problem is to get an approximation without a calculator. The video never got to it at all.
(-4)¹⁶=16,777,216
I don't know why YT recommended me this and I'm not a science major, but this intrigued me so much lol.
I went through my YT feed and marked all the politics, sports, pop culture, and crime stories as "not interested", and now finally I'm seeing interesting content like this in my feed. What took me so long?😀
@@richatlarge462I get not being interested in pop culture, sports and stuff, but why did you get rid of politics? Politics are highly important. Its what governs us and bad politics will have bad consequences in society and in people's lifes. When people aren't interested in politics, you eventually get leadership like Russia. So please take an interest in it. I'm sure you make use of your right to vote.
I really liked your explanation. Even tho I am out of maths for years now, with some thinking I was at least able to understand your steps!
Please DO NOT let RUclips sabotage your otherwise excellent videos with nested ads. They throw off the viewers' concentration.
This guy really loves this question
I appreciate that you don't waste time while giving presentation. 👌🏻
The math was easy, you holding two different colored markers with one hand and writing accurately, is what broke my brain
Nice problem.
Here's the Sagemath code (using Newton's method):
sage: reset()
sage: f(x)=x^(8-x)
sage: g(x)=diff(f(x),x)
sage: n(x)=x-g(x)/diff(g(x),x)
sage: x=4.0
sage: x=n(x);x
3.35700838162158
sage: x=n(x);x
3.53287101659463
sage: x=n(x);x
3.53539951720032
sage: x=n(x);x
3.53540103555960
sage: x=n(x);x
3.53540103556015
sage: x=n(x);x
3.53540103556015
sage: f(x)
280.904556712676
Note: I picked 4.0 as the starting value because it is between 0 and 8.
十分感謝老師解答😊😊
My approach before watching the video:
x^y is not defined for x0.
Since x^y < 1 for x>0 and y
Hmm this video demonstrates the difference in thinking between an engineer and a mathematician. I paused the video when the question was posed, opened excel, used one cell as an input for X, defined another cell as Y by making it 8 - the X cell, then defined a 3rd cell as the X cell to the power of the Y cell. At this point I used the solver function in excel to maximise the value of the 3rd cell by altering the value of the X input cell. Got the answer in less than a minute by brute forcing it 😅. I will admit blackpenredpen's method was more eloquent.
I'm right there with you. I might have done a column for x, another for 8-x and one more for the function. Range 3-5 with 0.05 increments, maybe refine around the maximum and called it good at 3.51.
Tried solving using desmos. Plotted x+y=8 first. Then plotted x^y = some constant b. The value of b where the two graphs only have one point of intersection is the maximum value of the function, and that came out to be approximately 280.902799
I have absolutely no ideia why RUclips recommend this video. Everything went WAAAAY over my head!
x+y=8, which also means x+(-y)= 8 (where x>y) or -x+y = 8 (where x
yes I know x&y should be >= 0...save that message..you're welcome..😊
x+y is no the same as x+(-y)
I really hate this question, let’s not take a look.
evil blackpenredpen be like:
@@yesiamrussian whitepencilgreenpencil
hey @blackpenredpen, can you make a shirt that shows a white and green pencil and says
"I really hate this question, let's not take a look
~whitepencilgreenpencil"? pweeeasee???
video ends
f(x) = x^(8-x)
ln(f(x))' = f'(x)/f(x)
the max is at f'(x) = 0, and f(x) can't be infinity so the max is at ln(f(x))' = - ln(x) + (8 - x)/x = 0
Ask a computer for an approximation
Instructions unclear. I now have 1,024 fishes.
After watching so many of your videos, I could easily find the answer by myself 😂 The lambert w function is like a normal function to me now.
I feel like we've been duped, he uses a black pen, red pen AND A BLUE PEN!!!
Seriously though the explanation made a lot of sense and the way you used three pens shows you're a pro at this lol.
This can also be solved using binary search
import math
low = 0
high = 8
delta = 1e-6
while abs(high-low) > delta:
mid = (low+high)/2
first = math.pow(mid-delta, 8-mid+delta)
second = math.pow(mid, 8-mid)
third = math.pow(mid+delta, 8-mid-delta)
if firstthird:
high = mid
else:
break
print(math.pow(low, 8-low))
Fun fact: if x+y=2e, the max of x^y is registered when x=y. In all other cases: if x^yy, if x^y>2e, max of x^y is reached when x
6:10 "Fishes" is correct here. It's a weird corner of English: it's "fishes" if they're easily countable, "fish" otherwise. No stranger than infinities, I guess.
Fishes is 'multiple fish of different species', if there are multiple fish of the same species, then they're fish. 6 fish, all salmon, vs 3 fishes, salmon, trout, mackerel.
@@Azmodon If you order fish for dinner, you might get 2 fishes on your plate. It's not just types of fish, it's simply countable vs uncountable. At least, per my OED Style Guide.
It's one of those funky words that is really dependent on the article used in front of it like the fishes vs fish like you were saying on it denoting a type but nothing more it seems
Literally the video I watched before this was a video on the W function on your shirt..
And now I see why.. this video is also about the function.. algorithm got me
Brillant explanation, I was able to follow along completely!
he really wanted to make sure we are aware of his appreciation for the question
As others have pointed out, you could use the same method to get x=k/W(k*e) for the max of x^y when x+y=k. The case of k=8 is interesting though, because 4^4 and 3^5 are so close together. We were already pretty close with 3.5^4.5.
My youngest son hit his high school math teacher with a similar answer as there was no criteria on a range for the problem, as with her and must all be positive.
Can you do an all-in-one calc question again? But please do a d/dx this time and put integrals inside
Edit: and i know he did it 4wk ago but i want one with a derivative
One fish - two fish, one sheep - two sheep, one buffalo - two buffalo
Quick education on fish or fishes: a fish is one singular fish, the fish are multiple fish from the same species of fish, the fishes are multiple fish from multiple species of fish :)
Notice that going from X^(8-x) to e^(8-x)lnx is valid if and only if x is > 0 ( can't be = to 0)
I got pretty far into the question before I eventually got stuck... Of course the Lambert W function was involved lmao, I still dont know how to properly use that thing
It's right there on his shirt.
(i'm kidding)
This is FASCINATING. You are a Master my friend 👏👏👏👏👏👏👏👏
I like how he shows that ChatGPT is unable to find the solution at the end.
Sure, it can spit out an explanation for how to find the solution, because those sorts of explanations are in its training data, but it can't process the solutions for itself because it doesn't "understand" what the explanation means. It can't create a mathematical model, much less perform the operations to process it, because it has no understanding or procedure for that. It only recognizes that this sort of problem resembles the maxima/minima problems from its training data, and regurgitates a procedure.
It's useful to recognize the limitations of the language model.
I don't pay for v4 but v3.5 has always been awful at calculus and has only gotten worse as they have limited the product. I love learning how to solve problems (hence why I am here) but I have yet to find a good AI solution.
OK, I did not expect the word FISH to be stuck in my head after a maths video.
with the constraints given where x can be a negative integer, the highest is incalculable ( infinity )
Here wrote a little python script don’t know if this will help
max_num = 0;
x = 1
y = 7
while (int(x+y) == 8):
print(x+y)
if (max_num < x**y):
max_num = x**y;
x+=0.0001
y-=0.0001
print(x+y)
print(max_num)
6:13 I didn't understood how did fish e^ fish is fish. And you said it is in calculus 1,but I have not heard of it
that was just another beautiful solution by my fav youtuber.
I really like this video.
I really like this video, let's have a look.
I really like this video, let's have a look!
Much more honest than other youtubers who cut every 2 seconds...
He might really love this question
Solved it with Lagrange multipliers. Glad I still remember Calculus III.
amazing explanation.❤
this was insanely epic, like for real, this was amazing and so fricking cool :>
It may be because I never studied it so I don't know much about it, but I don't really like the Lambert function. Since there's no way to determine its value without using wolpharm alpha or a software, it's not really explicitating or finding the value of X. I feel it's just writing an implicit equation in a different way that's still an implicit equation. It's like saying "there's no solution but here's one that isn't really one".
That's an inherent problem with inverses. Think of it like roots and logs where there's no other way to express those numbers. For example, √2 and ln(2) have no nice ways of expressing them, and you generally need a calculator to compute their values. Not much you can do about it when you have an irrational answer.
@@sethb124 but at least with square roots you can have a general idea of their value, you can at least figure out a min/max borders, like, you know that square root of 2 is somewhere between 1 and 2, you can, like, try out 1,1 * 1,1 then 1,2 * 1,2 and so on and figure out a few decimals this way.
None of that with the Lambert function. Either you leave it at that or use a software. At least as far as I know, there's no way to even figure out its value, better off go back to the implicit expression and try out values there.
@AcaciaAvenue Yeah, that's true. It's not super easy to estimate, but for problems like these, it's the most "exact" answer you'll get. It's not very intuitive, but I'd argue neither are lnx and e^x. It's still a valid answer because there are ways of estimating it (or else computers couldn't do it), it's just not an intuitive function.
@@sethb124That's the point I wanted to make. It's not an "exact" answer to me if I have no means of giving even an estimate numerical value. It's just picking something with no answer and writing it differently, but still with no answer.
If there were some means to calculate an estimate value of the lambert function without feeding it to a software and tell it "do it for me", then I will change my mind but I looked it up but there seems to be none. Therefore I don't see the utility of the Lambert function.
Totally agree. Just seems like a fix to me. And calling it exact is a stretch too. I solved this by implicit differentiation and Newton-Raphson. Didn't take long at all.
Thanks!
Ahh, Lambert W. I had to use it to solve an optimal control problem for landing a spaceship once.
Make it so, Ensign Cainghorn.
Much the same - I don't know if it is easier - but from the differential...
-ln x -1 +8/x = 0
ln x + 1 = 8/x
e.x = e^(8/x)
e.8 = (8/x)e^(8/x)
W(e.8)=2.26282674=8/x
x=3.535401
x^(8-x)=280.9045567
if X + Y = 8 then Y = - x +8. That looks like slope intercept form for a line that goes to infinity. So the answer is infinity^ infinity for max X ^Y.
duh:
without excluding negative numbers: (-infinity)^(infinity+8)
because both are even, it gives a positive answer which is infinitely large, thus providing a max that has no end
i used desmos for non whole numbers, and i used this equation:
y = x^(8-x)
I got 3.535^(8 - 3.535), which is approximately 280, which is greater than 4^4, or 256.
A math symbol I didn't know existed: 🐟(e^🐟)
I think if you wanted another number other than 8, all you need to do is replace 8 in the solutions with that number so get the answer
It's out of the scope of ChatGPT's algorithms LMAO
chat gpt just take info from the internet/database and provide an interpretation. if the database doesnt have the answer, chatgpt wont have it. it doesn't "think" on its own.
edit: actually it DOES learn, for it's an AI, but it specializes in predicting what the response to your question is base on the database of information it's accessing. But i dont think it can do math.
@@kingpetunfortunately chatgpt doesn't have the capabilities to do calculus but it might be able to do basic arithmetic, they could integrate something like Wolfram alpha into chatgpt one day
@@maddenbanh8033chatgpt has emergent problem solving as a result of understanding human language. Gpt-4 is able to use that problem solving to load arithmetic into premade calculators, like programming languages or Wolfram, in order to “solve” calculus and other math problems. It figures out what steps are necessary, and loads those steps into something that can actually get a confident result. Then it takes that result and moves on to the next step
I'm glad that I was able to solve it by myself. I found the maximum value of x by using newton's method directly as opposed to the lambert w function though (I vaguely recall that you find the values of W using newton's method anyway right?) .
This really reminded me how I hate this weird way to write 1, so confusing. And using chained fractions is ugly and unnecessarily confusing. And writing e^1 instead of e is curious.
Another Chatgpt response when I asked it to solve using calculus:
Certainly! To solve this problem using calculus, we first express \(y\) in terms of \(x\) using the constraint \(x + y = 8\), which gives us \(y = 8 - x\). Our function to maximize is therefore \(f(x) = x^{8 - x}\).
To find the maximum value, we need to:
1. Take the derivative of \(f(x)\) with respect to \(x\).
2. Set this derivative equal to zero and solve for \(x\) to find critical points.
3. Check these critical points to determine which one gives the maximum value of \(f(x)\).
The derivative of \(f(x)\) can be challenging because it involves both the power rule and the chain rule. Let's calculate it.
from sympy import symbols, diff, solve
# Define the symbol x
x = symbols('x')
# Define the function f(x) = x^(8 - x)
f = x ** (8 - x)
# Take the derivative of f with respect to x
f_prime = diff(f, x)
# Find the critical points by solving f'(x) = 0
critical_points = solve(f_prime, x)
# Show the derivative and critical points
f_prime, critical_points
The critical point for maximizing the function \(f(x) = x^{8 - x}\) is \(x = \exp(-1 + \text{LambertW}(8e))\), and the maximum value of \(f(x)\) at this point is approximately 280.905.
This result confirms our earlier graphical analysis, demonstrating that calculus provides a precise way to determine the maximum value of \(x^y\) given the constraints \(x, y \geq 0\) and \(x + y = 8\).
All of this, just to get .2 higher value.
But hey! Progress is progress! XD
Keep up the content! You've earned a sub!
I watched this guy because i needed help with dividing fractions or something like that, i have no idea whats going on but im watching anyways
the question gets pretty easy using Lagrange multipliers, but you'll end with a x(lnx + 1) = 8.... and then i got stuck, thanks god to W(.) function
when he started demonstrating functions with fish i thought i was hallucinating
I think he really likes this question
I love how all his fish are evil
Thank you,! I wasted some time trying to solve this only with derivates and found only (0, 8) and (8,0) 😞.
I don't even remember the Lambert W from school. Very nice!
Well, do we assume x and y positive?
x=-50 y=58 still produces an x+y=8, meaning any arbitrarily big and even negative x will solve to a greater arbitrarily big value of x^y.
Are there solutions if X and Y are complex numbers?
No, because you cannot compare the size of imaginary numbers. You would have to rephrase to find the maximum of |x^y| for instance.
Complex numbers arent ordered bud
or if you're me and don't know calc but you know python and are a fan of bruteforcing, you write this lol
def max(iteration_amt, sum_amt):
i, x, y, max_found = 0, 0, 0, 0
while i < sum_amt:
x = sum_amt - i
y = sum_amt - x
if x**y > max_found:
max_found = x**y
best_nums = [x, y]
i += iteration_amt
return max_found, best_nums
print(max(0.001, 8))
(decrease iteration amount for higher decimal precision)
Over compiling 4+4. Variables are exactly that and anything can be done with them.
My first thought was x=-1000, y=1008 works so there's no bounds to the ultimate max value of x^^y.
there is no maximum: for any pair number y, we have (8-y)^y as big as possible.
example y=1008 x=-1000 , -1000^1008 = 10^3024...
I don't know, I just have the feeling...but I'm pretty sure he loves the question
Two iterations of Python got me the answer: "for f in [x * .01 for x in range(340,370)]: print(f, f**(8.0-f))"