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

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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[@blackpenredpen]

all solutions to 2^x-3x-1=0 (transcendental equation) kzbin.info/www/bejne/fXvFq6ajmMh5qtk

[@pokemonjourneysfan5925]

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]

🎉🎉🎉

[@rpocc]

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]

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

[@cabbycabbycabbycabbycabby]

(-4)¹⁶=16,777,216

[@farhansadik5423]

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

[@user-zs3st5qq6r]

Yeah, I really like this question

[@geraldpujaya6682]

he really likes the question

[@jamescollier3]

7:56 you are

[@rodrigoappendino]

I thought he was glitching.

[@EgzoDiuse]

i wonder if he likes the question

[@hysterixation]

"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]

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

[@giacomocasartelli5503]

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

[@the_pks_]

where are his other videos with that start lol

[@Dibat24]

I love this question too!

[@CCheukKa]

He really loves this question

[@user-to1xe7rt4j]

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

[@Friendly-Neighborhood-Asexual]

😂😂😂

[@mayankdave9739]

I see what you did there

[@Nico2718_]

More like fish*e^fish 😂

[@electricgamer_yt4753]

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

[@la.zanmal.]

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

[@elimondingwiza3168]

Elite comment. Simply beautiful

[@joshavery]

Underrated comment

[@anonymouschessaccount5508]

W COMMENT THIS DESERVES TO BE POPULAR THE creativity omg

[@anonymouschessaccount5508]

🎉

[@GoofyGangster-bo4zn]

This is so underrated

[@SabertoothDeathmouse]

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]

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]

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

[@jay-5061]

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

[@HienNguyen-cs1md]

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

[@battle00333]

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]

Isn't that common sense lol

[@michaelleue7594]

Or more directly, (1024)^2.4.

[@trueriver1950]

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]

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]

​@@xlr8_bs514ikr

[@vascomanteigas9433]

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

[@subjekt5577]

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

[@bleaks218]

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]

I thought of this method too!

[@meurdesoifphilippe5405]

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

[@Enlightenment0172]

(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]

​@@Enlightenment0172!??

[@TheSkullkid16]

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]

I was thinking just that!

[@AzureKyle]

@@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.]

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]

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]

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

[@FourOneNineOneFourOne]

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]

@@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]

@@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]

@@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]

@@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]

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]

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

[@adammizaushev]

@@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]

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]

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

[@abrahamholleran4162]

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

[@frimi8593]

@@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]

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

[@vez3834]

​@@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.)

[@UENShanix]

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

[@bigmiraclewhips]

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

[@xavier9577]

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

[@electricnezumi]

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

[@Shankster49]

yeah this guy is using calc 1 and maybe calc 2

[@nurajannattaslima9853]

pretty straight forward

[@neilmorrone691]

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!

[@einargs]

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

[@matteo2042]

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!

[@trueriver1950]

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]

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]

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]

@@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]

@@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]

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.

[@tobiasruck]

Brillant explanation, I was able to follow along completely!

[@bananasplitd006a5]

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

[@thatssokwekwe]

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

[@tomasdvorak7307]

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

[@ricardcalonge4542]

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

[@BobChess]

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]

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

[@AxillaryPower2]

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)

[@user-gw7tu4nf4u]

十分感謝老師解答😊😊

[@andrewhalyburton5990]

This guy really loves this question

[@kafureszipuskza3834]

amazing explanation.❤

[@TheScotty1701d]

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.

[@SebastianGarcia-qo7wi]

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

[@savitsios]

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

[@zhabiboss]

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

[@ronbannon]

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.

[@alanhersch4617]

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?

[@omaanshkaushal3522]

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

[@tyreathian]

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

[@mike1024.]

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]

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

[@bernardcrnkovic3769]

such a satisfying answer!

[@matthewTaylor1990]

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]

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.

[@louisng114]

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

[@TheProGamerMC20]

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.

[@NagabhushanBaddi]

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))

[@phicrow]

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?) .

[@gaspi91]

That marker switches were smooth 👍

[@aeschynanthus_sp]

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

[@davestorm6718]

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

[@edward_lado]

He might really love this question

[@otbwwilliams]

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

[@CrimsonHexx]

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]

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

[@makisekuurisuu]

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

[@kingpet]

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]

​​@@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]

@@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

[@Edzward]

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

[@patrickrobertshaw7020]

How would you solve Lambert W by hand? This seems extremely hand wavey by just invoking an inverse of what feels arbitrary, and then asking wolfram alpha to actually solve it. How would this be any different than just asking wolfram to do it from the beginning? Does W have an alternative definition that allows this to actually be calculated reasonably?

[@ripayanami]

No easy way, that's why it was not used till 1980 despite being found in 1780

[@GK-gc9cv]

Basically it's xe^x = 8e where x = lnx + 1. Solving xe^x isnt trivial, but solving lnx + 1 is. So yes Basically a calculator that could solve xe^x = number would be able to solve the original equation, you need like a TI89 to do it

[@a.tsuruya8]

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

[@omuploads]

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

[@Peter-qv6ke]

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

[@user-is9wb6de6c]

Thanks to you I discovered the Lambert W function.

[@RubyPiec]

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]

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]

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

[@GK-gc9cv]

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]

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.

[@SuperDeadparrot]

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?

[@boxvism]

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!

[@SkorjOlafsen]

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.

[@calcu_]

I just have a question about the e expression. You said, and it's true that x^(8-x) can be written as e^((8-x)ln(x)) but there is something I can't figure out. For x in R, e^x > 0 this is a rule. But, when x=0, x^(8-x) = 0 Which is not true for the other one, so by doing this transformation we lost one of the definitions of the original one? Doesn't that affect the demonstration?

[@olixx1213]

ln(x) has no value when x is 0

[@calcu_]

@@olixx1213 yes I know

[@Samuelllllllllll]

in R, x^(8-x) can be written as e^((8-x)ln(x)) only if x^(8-x) > 0, because one step in this process involves ln(x^(8-x)). so this equality doesn't hold when x^(8-x) = 0 (which happens when x=0). for this demonstration, since we know x≥0, we can handle the x=0 case first then proceed with the rest knowing x>0 and that writing it as e^((8-x)ln(x)) would be valid

[@calcu_]

@@Samuelllllllllll yes thank you, that's what I thought. I just wanted to make sure

[@SnijtraM]

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

[@artofsalsa5268]

Would the lambert function work with a dolphin? Because strictly speaking a dolphin is a mammal, not a fish.

[@egodreas]

Mammals are not a problem. It even works with unicorns, hippogriffs, and other animals with imaginary components.

[@poket5560]

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)

[@thesparksplug]

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.

[@rskne8803]

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]

I think you commented on a wrong video?

[@rskne8803]

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

[@candy-coatedcloud]

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

[@Bv-yl5dg]

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

[@SJrad]

How is the lambert function of something like 8e calculated. It just feels like something black box-y

[@Qeisama]

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

[@richatlarge462]

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]

​@@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.

[@user-tw8ki1jz1r]

looked for math, found numerical methods 😢

[@Manushen]

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

[@Manushen]

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

[@Reggat]

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

[@SidneiMV]

Amazing question!

[@windowsforvista]

I never heard of W before, very cool

[@nielsniels5008]

I loved this solution

[@sigmascrub]

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]

Then apply the Newton-Raphson method to solve for x

[@GrandRezero]

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

[@GrandRezero]

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

[@JonnyCakesFCB]

Great video!

[@qzwxecrv0192837465]

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.

[@DrMcFly28]

This was fishier than I expected

[@cainghorn]

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

[@gregnixon1296]

Make it so, Ensign Cainghorn.

[@nnoxie.a]

i just looked at the thumbnail and before i knew that it was for non-negative values i thought of something like (-X)^(8+X) where X is any large even number.

[@ahmedelakad8875]

Can we use lagrange multipliers?

[@gabest4]

Nice, now let's see the solution by approaching the problem from (8-y)^y

[@OrdinarySonicfanMmKay]

"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

[@kemcolian2001]

IDK why, but i am obsessed with BPRP's Lambert W function Videos

[@deltalima6703]

Any complex solutions?

[@LycorisLilyP]

I wish that my professors were as prepared w/ markers as you are in the back 😭 can't ever see what my profs are writing

[@user-bq5vq6zq8w]

Hello, what I'm wondering what if we use the riman-liuville differintegral and made with it some surfaces where one variable is a order of differintegral and other one is a argument of a fuction, how it's will look like P.S: sorry for my bad english ¶:

[@SirNobleIZH]

I love how all his fish are evil

[@pajeetsingh]

This is very numerical and algebraic way of solving this problem.

[@VladimirPutin21172]

Can you please solve for a b and c the equation 4×a^2+4×b^2+3=4c+4 sqrt of a+b-c . Is for a friend.

[@MrAustindanderson]

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

[@MrMooooole]

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.

[@RavenMobile]

Before clicking on the video (for the second time, forgot the answer from the first time I watched it), I decided that the largest would be a decimal number... if raising to a non-integer power was actually possible (hadn't seen that before). But, I had no clue how to do any of the math to actually figure it out, other than perhaps trying to plug it into a graphing calculator and stepping along the curve to see the highest part.