if x+y=8, find the max of x^y

Рет қаралды 689,095

5 ай бұрын

If x+y=8 for nonnegative x and y, then find the maximum value of x^y. This question seems like a typical calculus optimization problem, but to find the exact answer, we will need to use the Lambert W function. • Lambert W Function (do...
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Пікірлер: 747

@blackpenredpen 5 ай бұрын

all solutions to 2^x-3x-1=0 (transcendental equation) kzfaq.info/get/bejne/fbCSrdaeyMmqpKs.html

@pokemonjourneysfan5925 5 ай бұрын

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?

@jonathanv.hoffmann3089 5 ай бұрын

🎉🎉🎉

@rpocc 5 ай бұрын

The finest value of X I’ve found is 3.53540103 but I have no clue how to express the limit of x.

@user-ig2hy9pz1g 3 ай бұрын

I think the essence of this problem is to get an approximation without a calculator. The video never got to it at all.

@cabbycabbycabbycabbycabby 2 ай бұрын

(-4)¹⁶=16,777,216

@farhansadik5423 5 ай бұрын

I literally thought I was glitching with how many times I heard him say 'I really like this question" lmao

@user-zs3st5qq6r 5 ай бұрын

Yeah, I really like this question

@geraldpujaya6682 5 ай бұрын

he really likes the question

@jamescollier3 5 ай бұрын

7:56 you are

@rodrigoappendino 5 ай бұрын

I thought he was glitching.

@EgzoDiuse 5 ай бұрын

i wonder if he likes the question

@hysterixation 5 ай бұрын

"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

@codycast 3 ай бұрын

It’s a dude that holds a little lapel mic. I don’t think he’s too concerned with edit quality.

@giacomocasartelli5503 3 ай бұрын

I don't know about editing, but I think he really loves this question

@the_pks_ 2 ай бұрын

where are his other videos with that start lol

@Dibat24 Ай бұрын

I love this question too!

@CCheukKa 29 күн бұрын

He really loves this question

@user-to1xe7rt4j 5 ай бұрын

lambert W function be like : input→fishy fish output→fish

@Friendly-Neighborhood-Asexual 3 ай бұрын

😂😂😂

@mayankdave9739 3 ай бұрын

I see what you did there

@Nico2718_ 2 ай бұрын

More like fish*e^fish 😂

@electricgamer_yt4753 Ай бұрын

BPRP’s ‘lil fish drawings are so cute. Fish*e^fish return fish for life.

@la.zanmal. 5 ай бұрын

See, the mistake at the end is that you asked ChatGPT 3.5, you should have asked ChatGPT 8/W(8e).

@elimondingwiza3168 4 ай бұрын

Elite comment. Simply beautiful

@joshavery Ай бұрын

Underrated comment

@anonymouschessaccount5508 5 күн бұрын

W COMMENT THIS DESERVES TO BE POPULAR THE creativity omg

@anonymouschessaccount5508 5 күн бұрын

🎉

@SabertoothDeathmouse 5 ай бұрын

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?"

@sfglim5341 5 ай бұрын

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

@jack002tuber 5 ай бұрын

@@sfglim5341 I understood it all till we hit that lambert W thing. I have never seen it before

@jay-5061 4 ай бұрын

@@jack002tuberits a very handy tool in college/uni algebra and calculus.

@HienNguyen-cs1md 4 ай бұрын

@@jack002tuber It's just a function, you don't need to delve into it too much

@battle00333 5 ай бұрын

At the very start for (-4)^12 it is the same as (-2)^24, for anyone wondering just how much bigger it is.

@xlr8_bs514 5 ай бұрын

Isn't that common sense lol

@michaelleue7594 5 ай бұрын

Or more directly, (1024)^2.4.

@trueriver1950 5 ай бұрын

Which is slightly more than 16 million, because 2⁴ = 16, and 2²⁰ is 1024² > 1,000² So yeah, I'm not arguing with "bigger" 😉

@nunosantos4217 5 ай бұрын

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

@brahmbandyopadhyay 5 ай бұрын

@@xlr8_bs514ikr

@vascomanteigas9433 5 ай бұрын

Using Lagrange multipliers should solve the problem. x=1/lambert_w(1) and y=8-1/lambert_w(1)

@subjekt5577 3 ай бұрын

for the ignorant, forgetful, and lazy like myself: en.wikipedia.org/wiki/Lagrange_multiplier

@bleaks218 2 ай бұрын

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.

@keaganlove1478 Ай бұрын

I thought of this method too!

@meurdesoifphilippe5405 5 ай бұрын

I did not notice that x and y should be nonnegative, else you could have (-2k)^{8+2k} which goes to infinity.

@Enlightenment0172 5 ай бұрын

(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.

@OnlyTheQuack 5 ай бұрын

@@Enlightenment0172!??

@TheSkullkid16 5 ай бұрын

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.

@jakeklic 5 ай бұрын

I was thinking just that!

@AzureKyle 5 ай бұрын

@@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.

@mike1024. 5 ай бұрын

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.

@frostiifae 4 ай бұрын

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.

@atriyakoller136 5 ай бұрын

As a non-mathematician I understood nothing but the way you explain is so cool that it made me more enthusiastic for maths😊

@FourOneNineOneFourOne 5 ай бұрын

as non-mathematician you should still know highschool maths, unless you're still in school, then I guess you'll still learn about this.

@atriyakoller136 5 ай бұрын

@@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.

@apljack 5 ай бұрын

@@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.

@JonesyTerp1 5 ай бұрын

@@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.

@GFlCh 3 ай бұрын

@@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.

@adammizaushev 5 ай бұрын

Considering x >= 0, y >= 0 on my own, I got stuck with x(ln(x) + 1) = 8 Thank you for introducing the W function

@TomJones-tx7pb 2 ай бұрын

yeah but you already had a function, so rephrasing it as another function is not really solving anything.

@adammizaushev 2 ай бұрын

@@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

@herardpique7302 5 ай бұрын

The most fascinating about this, is if you do this with X + Y = 100, the maximum will be 24^76, which is totally asymmetrical

@frimi8593 5 ай бұрын

You mean to tell me that e^{W(100e)-1} is equal to 24?

@abrahamholleran4162 5 ай бұрын

Close - the x is still e^(productlog(100e)-1) or 23.947

@frimi8593 5 ай бұрын

@@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

@gregorymorse8423 4 ай бұрын

W(e)=1 so it's a shame there aren't identities for cases a constant precedes e.

@vez3834 3 ай бұрын

@@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.)

@bigmiraclewhips 2 ай бұрын

didnt understand a word of this but i am now holding a W and a bunch of fish

@UENShanix 5 ай бұрын

This is a great video and you are a wonderful mathematical communicator! Thank you for creating this!

@xavier9577 25 күн бұрын

I really hate this question, let’s not take a look.

@nurajannattaslima9853 5 ай бұрын

pretty straight forward

@trueriver1950 5 ай бұрын

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 ?😊

@carryingautoclicks7501 5 ай бұрын

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.

@KenFullman 5 ай бұрын

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.

@AzureKyle 5 ай бұрын

@@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.

@Tzizenorec 5 ай бұрын

@@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. :)

@ripayanami 3 ай бұрын

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.

@neilmorrone691 5 ай бұрын

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!

@electricnezumi 4 ай бұрын

I often find your videos a bit beyond my skill level but I found this one really approachable and fun. thanks for this!

@Shankster49 4 ай бұрын

yeah this guy is using calc 1 and maybe calc 2

@matteo2042 5 ай бұрын

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!

@einargs 5 ай бұрын

This is super fun to watch. I had no idea about the W function.

@BobChess 5 ай бұрын

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!

@MozartEstLa 5 ай бұрын

Much more honest than other youtubers who cut every 2 seconds...

@tobiasruck 5 ай бұрын

Brillant explanation, I was able to follow along completely!

@bananasplitd006a5 2 ай бұрын

I'm happy to say that this is the first problem from your videos that I have been able to solve !🎉

@andrewhalyburton5990 26 күн бұрын

This guy really loves this question

@thatssokwekwe 5 ай бұрын

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

@ronbannon 5 ай бұрын

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.

@ricardcalonge4542 5 ай бұрын

This is FASCINATING. You are a Master my friend 👏👏👏👏👏👏👏👏

@tomasdvorak7307 5 ай бұрын

I appreciate that you don't waste time while giving presentation. 👌🏻

@user-gw7tu4nf4u 5 ай бұрын

十分感謝老師解答😊😊

@kafureszipuskza3834 5 ай бұрын

amazing explanation.❤

@otbwwilliams 2 күн бұрын

I don't know, I just have the feeling...but I'm pretty sure he loves the question

@phicrow 5 ай бұрын

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?) .

@makisekuurisuu 5 ай бұрын

It's out of the scope of ChatGPT's algorithms LMAO

@kingpet 4 ай бұрын

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.

@maddenbanh8033 3 ай бұрын

@@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

@prismglider5922 2 ай бұрын

@@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

@TheScotty1701d 4 ай бұрын

Quite interesting. I didn't know the Lambert W function and ended up with (8-x)/x - ln x = 0. I found the approximation using my almost 40y old pocket calculator by transforming this to x = 8/(ln x + 1) and iterating with the initial value x0=4. Convergence isn't very fast, but at the end of the day you have to solve this by using some form of iteration. Nevertheless a really interesting function.

@bernardcrnkovic3769 5 ай бұрын

such a satisfying answer!

@savitsios 5 ай бұрын

I'm a computer scientist who hasn't done math in a couple of years, it felt good to get this on my own 😌

@SebastianGarcia-qo7wi 5 ай бұрын

Solved it with Lagrange multipliers. Glad I still remember Calculus III.

@user-tw8ki1jz1r 2 ай бұрын

looked for math, found numerical methods 😢

@omaanshkaushal3522 13 күн бұрын

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

@Bv-yl5dg 26 күн бұрын

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

@JonnyCakesFCB 4 ай бұрын

Great video!

@edward_lado 5 ай бұрын

He might really love this question

@CrimsonHexx 5 ай бұрын

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

@EmpyreanLightASMR 5 ай бұрын

It's right there on his shirt. (i'm kidding)

@imincent1767 29 күн бұрын

I think he really likes this question

@Manushen 2 ай бұрын

x+y=8, which also means x+(-y)= 8 (where x>y) or -x+y = 8 (where x

@Manushen 2 ай бұрын

yes I know x&y should be >= 0...save that message..you're welcome..😊

@gaspi91 4 ай бұрын

That marker switches were smooth 👍

@SidneiMV 5 ай бұрын

Amazing question!

@Reggat 5 ай бұрын

All of this, just to get .2 higher value. But hey! Progress is progress! XD Keep up the content! You've earned a sub!

@alanhersch4617 5 ай бұрын

If I understand this right, this means we now have a generic formula for any number x + y has to add up to right? Because you can just replace the 8 with whatever the sum is?

@MrMooooole 3 ай бұрын

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.

@windowsforvista 5 ай бұрын

I never heard of W before, very cool

@SkorjOlafsen 5 ай бұрын

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.

@zhabiboss 5 ай бұрын

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

@matthewTaylor1990 5 ай бұрын

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.

@bertblankenstein3738 Ай бұрын

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.

@AxillaryPower2 2 ай бұрын

What course teaches the Lambert W function? I'm pretty sure I didn't see it in my high school calculus class (although it was 16 years ago so maybe I forgot)

@boxvism 2 ай бұрын

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!

@tyreathian 5 ай бұрын

Could you make a video on how to find the points where a polar curve's tangent lines are at a given slope?

@mike1024. 5 ай бұрын

You need the rectangular formula for a polar derivative. Do a quick Google search to find it: "dy/dx for polar function" (too complicated to try to write the formula here). Once you have that, you can set it equal to any slope you want and solve. You might get multiple ways to express the same point though.

@tyreathian 5 ай бұрын

i know the formula but I was struggling it its usage@@mike1024.

@user-is9wb6de6c 4 ай бұрын

Thanks to you I discovered the Lambert W function.

@TheProGamerMC20 3 ай бұрын

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.

@omuploads 5 ай бұрын

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

@sakuyarules 5 ай бұрын

That was really cool.

@aeschynanthus_sp 3 ай бұрын

About using fish in a function: my maths teached in high school used horse heads similarly!

@Peter-qv6ke 5 ай бұрын

ChatGPT can’t even exceed calculus teacher That’s what I like to see❤

@rskne8803 3 ай бұрын

We just covered that a couple of months ago. Wouldn’t it be easier to just substitute y = kx and get kx/x = k => limit depends on the slope k, therefore it doesn’t exist?

@blackpenredpen 3 ай бұрын

I think you commented on a wrong video?

@rskne8803 3 ай бұрын

@@blackpenredpen Oh wow. That's weird. Yeah, something was wrong with the API, I guess.

@IoT_ 5 ай бұрын

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 \(x + y = 8\).

@NagabhushanBaddi 4 ай бұрын

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))

@poket5560 5 ай бұрын

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)

@SnijtraM 3 ай бұрын

A math symbol I didn't know existed: 🐟(e^🐟)

@GrandRezero 3 күн бұрын

Literally the video I watched before this was a video on the W function on your shirt..

@GrandRezero 3 күн бұрын

And now I see why.. this video is also about the function.. algorithm got me

@cainghorn 5 ай бұрын

Ahh, Lambert W. I had to use it to solve an optimal control problem for landing a spaceship once.

@gregnixon1296 5 ай бұрын

Make it so, Ensign Cainghorn.

@nielsniels5008 5 ай бұрын

I loved this solution

@SuperDeadparrot 3 ай бұрын

I haven’t done the calculation, but could you not have set F(x,y) = x^y - lambda * ( x + y ) and then minimized this function by eliminating the lambda parameter?

@SirNobleIZH 5 ай бұрын

I love how all his fish are evil

@OrdinarySonicfanMmKay 5 ай бұрын

"I really like this question, let's h- I really like this question, let's have a look. I REALLY LIKE THIS QUESTION, LET'S HAVE A LOOK." - blackpenrenpen 2023

@davestorm6718 3 ай бұрын

with the constraints given where x can be a negative integer, the highest is incalculable ( infinity )

@EmpyreanLightASMR 5 ай бұрын

I like how it sounds like you say "ys" and the W(8e) in your handwriting looks like the word "WISE" to me. So it all came round full circle 😂 As a non-mathematician, while it's impressive to see someone figuring this out, I'm more in awe that someone figured out the methods needed to figure this out first!

@MrFirelord 4 ай бұрын

Thanks!

@MrAustindanderson 2 ай бұрын

Cool and all, but a result with the productlog function in it is just as far away from a usable decimal format as entering solve(derive(x**(8-x),x)=0,x) if you need wolfram or another CAS to evaluate

@louisng114 5 ай бұрын

You can also solve it by setting its partial with x to its partial with y.

@anonymouscheesepie3768 5 ай бұрын

love this!

@gownerjones1450 5 ай бұрын

If you ask GPT-4 instead of GPT-3.5, you get the correct answer. ChatGPT will now write python programs to calculate precise results for questions like these.

@Hawking1969 4 ай бұрын

Very clever!

@herardpique7302 Ай бұрын

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

@thesparksplug 5 ай бұрын

I know I am really old when a function is introduced and becomes popular after I have finished university. 😂😂 I would love to know how the Lambert-W function functions - if it can be broken down into a series function like others.

@sigmascrub 5 ай бұрын

x+y=8 y=-x+8 Therefore, x^y=x^(-x+8) Take the first derivative and set it equal to zero ... that's as far as I got 😅

@vanshjain3428 5 ай бұрын

Then apply the Newton-Raphson method to solve for x

@a.tsuruya8 5 ай бұрын

OK, I did not expect the word FISH to be stuck in my head after a maths video.

@RubyPiec 5 ай бұрын

is it possible to calculate W(n) with a regular calculator? or do you need one with a specific button for the lambert w function?

@ryanburkett949 5 ай бұрын

With a TI-84 or the like you could do the integral definition and use it. It wouldn't be super quick but it would work. Could write a program on it to do it for you.

@RubyPiec 5 ай бұрын

@@ryanburkett949 but nothing with an old calculator like a casio fx-82ex they make me use at school?

@GK-gc9cv 2 ай бұрын

Just graph xe^x = 8e and you solve it. Since once you get x then set that = ln x + 1 which you can solve by hand

@bertblankenstein3738 Ай бұрын

If you are going to program a calculator, you might as well skip W and go straight to x^(8-x). Much more direct. You won't get the W but still get the answer.

@Edzward 5 ай бұрын

I have absolutely no ideia why KZfaq recommend this video. Everything went WAAAAY over my head!

@Qeisama 5 ай бұрын

I don't know why YT recommended me this and I'm not a science major, but this intrigued me so much lol.

@richatlarge462 2 ай бұрын

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?😀

@david4649 2 ай бұрын

@@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.

@spideybra 12 күн бұрын

When you get P(x) = e^((8-x)lnx), can't you assume that max P => max (8-x)lnx and derive that instead? Then you don't have to deal with the exponential.

@maxxme6746 5 ай бұрын

One fish - two fish, one sheep - two sheep, one buffalo - two buffalo

@chickens_are_sus6231 10 күн бұрын

I think he really liked that question

@DrMcFly28 Ай бұрын

This was fishier than I expected

@deltalima6703 5 ай бұрын

Any complex solutions?

@MrFrmartin 4 ай бұрын

He really really really loves this question

@candy-coatedcloud 2 ай бұрын

I first calculated the dy/dx of xraisedtoy and then I solved a differential equation, assumed the arbitrary constant to be 0 and then we get x=y, and hence option b from thumbnail ie 4 power 4