Lagrange Multipliers with TWO constraints | Multivariable Optimization

In our introduction to Lagrange Multipliers we looked at the geometric meaning and saw an example when our goal was to optimize a function (i.e. find maximums and minimums) subject to ONE constraint: • Lagrange Multipliers |...
Now we are upgrading to the case of optimizing with two constraints. We will look at how to interpret the lagrange multiplier method geometrically for two constrains, and then see a full example. We will also look at the geometry of the special case of optimization function: the distance.
Geogebra Link for the animations at the end: www.geogebra.org/classic/p3nn...
0:00 Intro
0:38 Lagrange Multiplier Method
4:50 Example
12:30 Visulization
Click Multivariable Calculus playlist below for the rest of the series.
****************************************************
COURSE PLAYLISTS:
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►DIFFERENTIAL EQUATIONS (Calc IV): • How to solve ODEs with...
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
***************************************************
► Want to learn math effectively? Check out my "Learning Math" Series:
• 5 Tips To Make Math Pr...
►Want some cool math? Check out my "Cool Math" Series:
• Cool Math Series
****************************************************
►Follow me on Twitter: / treforbazett
*****************************************************
This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: www.amazon.com/shop/treforbazett

[@SolaceAndBane]

The only guy on KZbin who gives an explanation for the expanded Lagrange Multiplier, even my professor just threw the formula out and told us to use it

[@atakan716]

i wish all classes were like this, all we need is just a touch of intuition and visualization to set the concepts clear in our mind!

[@tasninnewaz6790]

I love Trefor for Math and his personality.

[@joaomattos9271]

I've watched many classes on youtube and I can say that Professor Trefor's classes stand out. Simply awesome!

[@cesarnunezrios5595]

Dr. you are amazing! You just earned a new follower. This video really helped me

[@nguyentuanminh5330]

What an awesome explainations and cool visualization. Thanks you Prof, keep doing.

[@arinoba492]

Nice energy and even better teaching! I also found that website and seeing it here makes me happy :D

[@Speak4Yourself2]

Outstanding tutorial. Thanks a lot!

[@rajat4260]

Thanks man ...you just made my life easier...gr8 work..

[@josecarlosferreira4942]

Thank you man! You are very helpful =D

[@briandwi2504]

That was beautiful. I suddenly noticed while watching the video that I too was wearing a checked shirt! Morphing into Dr Trefor!

[@nirajgujarathi6796]

thanks professor, it is really great explanation !

[@abmohit3117]

Just amazing ❤️

[@ar3568row]

This course/playlist is extremely great , wish I found it earlier , now my exam is tomorrow itself 😕

[@grapplerart6331]

After reading your comment, I can infer why you didn't find it earlier.

[@ar3568row]

@@grapplerart6331 and yes, you are inferring correctly

[@grapplerart6331]

@@ar3568row 🤣🤣🤣 How did it go?

[@devashishshah9021]

Your explanation, math, handwriting, 3d graphs.... all are super good

[@devashishshah9021]

But your mu looks like a hook :-)

[@FranFerioli]

11:50 How satisfying when you catch the Professor making a clerical sign mistake. 11:58 How disappointing when such clerical sign mistake gets squared off leaving the correct result 😁 Great video as usual!

[@continnum_radhe-radhe]

Thanks a lot sir 🔥🔥🔥

[@milanrai6988]

Really really helpful for me

[@fernandojackson7207]

Excelent, as usual. Why not just find the intersection of the two constraints and use the standard method on that intersection?

[@Eric-gc3gh]

Thanks from Korea

[@user-lh5xs6xn9s]

Thank you sir you saved me.

[@schizoframia4874]

This is very cool

[@crimfan]

Zed's dead, baby, Zed's dead. ;) Nicely done. I really like the visualizations, too. I'll have to check out the software you mentioned in one of these vids.

[@user-og7lf9oe7x]

thank you so much

[@steveying1305]

great video

[@Lets_MakeItSimple]

Your channel is so underrated.

[@DrTrefor]

I appreciate that!

[@sdnavghare]

Thank you sir

[@vallurusudheerbabu8294]

i became a big fan to ur intention.

[@tiandao1chouqin]

Great video. But I'm wondering if there is a better explanation than just by extension of the 1 constraint case? If del f is orthogonal to both del g1 and del g2, then should del f be the cross product of the two?

[@NubaPrincigalli]

Thanks for the excelent content! Found a small typo: at 11:50, it should be f(-3,0,-3), not f(-3,0,3), as z was squared the typo went unnoticed.. :)

[@mathman2170]

Nice.

[@suhailawm]

tnx aLot prof.

[@Darkev77]

Genius

[@Saminnik]

hi, thank you for this video. I want to know if distance optimization is basically distance minimisation?

[@user-gc9hn8yw4d]

@3:03 "single variable function" -> single constraint case

[@srikanthk1276]

Hi Trefor, you made it look easy. Thank you👍 I didn't understand why grad f is a linear combination of gradients of the two constraints. Shouldn't grad f be perpendicular to the line of intersection of constraints? Can't one find the gradient of the intersection line and then proceed the same way as Lagrange multiplier case for a single constraint?

[@DrTrefor]

This is a fine method, but often finding a nice description for that line of intersection is highly non trivial

[@user-xw4lu9py9r]

The Graphical Approach in 3-D. You could possibly draw the surfaces by hand and compare the drawing to Geogebra. I tried this course with my 2-D calculator, and of course I could not visualize 3-D well.

[@user-zu3hc2jn4l]

really appreciate your lesson. i just finished the high school math lessons and didnt major in math when in college. i tried lagrange multipliers on the following question, but was still stocked. too hard to solve the equation. abc=23, ab+bc+ca=27, what is the max and min of a^2+b^2+c^2 really appreciate if you can help......thank you.

[@bzboii]

I have that shirt! Nice!

[@piotrfranczak8757]

Thanks

[@DrTrefor]

Thank you so much!

[@Aim4sixmeals]

Hi sir could u make videos on statistics? Like t tests, nullhypotheses

[@marcellozanardelli7713]

Nice video, but I didn't get how do you get to the linear combination of the gradients of the constraints? I get the 1 constraint case, but cannot understand the extension to this case

[@HermanToMath]

me neither. I understood gradient f and gradient g are parallel. However, if gradient g1 and gradient g2 form a plane, and gradient f is normal to the plane, it means gradient f is at the same time perpendicular to both gradients of g1 and g2.

[@youssefdirani]

@@HermanToMath no, the grad is perpendicular to both curves not to their grads

[@aashsyed1277]

what about 3 constraints and 4 variables? are the equations gonna be same with one more constraint and one more variable like delta? and at 9:51 why is the case not possible?

[@eldhopaul683]

Which software you used to generate all the animations? Is it Geogebra?

[@sayanjitb]

Wow, this kinda interpretation is pretty handsome. Dear sir I had a question, i have seen in some places circle is indicated as S1 and sphere in 3d as S2. What do they mean anyway? TIA

[@DrTrefor]

it is just shorthand for a "1 dimensional sphere" and a "2 dimensional sphere" and you could go further o an n dimensional sphere which are all he points in n+1 dimensions of equal length fro the origin.

[@abzshaker]

your videos are amazing but maybe you could get a better mic....and your channel would be perfect in all points!!

[@Han-ve8uh]

Im completely lost from the statement "the normal to g2 surface is gradient of g1". At 2:35 that vector really looks normal to g1 surface rather than g2, which is inconsistent with the audio saying that's the normal to g2? Is there a video in the playlist explaining this? My understanding of gradient vectors stopped at the "Geometric Meaning of the Gradient Vector", where it lives in the x-y plane and points to direction of steepest ascent on the surface. It seems that the gradients in this video do not stay flat on x-y plane. How can they be visualized and is there a video in playlist on gradients that don't live in just x-y plane and their geometric meaning? How would these g1 g2 and f gradients look on the geogebra example in last part of video? I wish the later example referred back to the theory in the first part of video.

[@KidsCastable]

same here, no clue how those words are true and confused by the pic not following the words nor the math I understand

[@dhyanprasad5611]

i wish you were my teacher

[@HelloThere-lo3qi]

but how to solve it if all the x, y,z equation has 2 variable constraints and some of em even has the xyz variable on it, hence i cant make the same solution like yours sir, since i cant assume anything T.T

[@ar3568row]

11:13 is it necessary that points we get with the help of Lagrange Multipliers , are either max or min . Why don't we consider the possibility of saddle point ?

[@robinbernardinis]

The intersection of the restraints is 1 dimensional, so there can't be saddle points. In general, though, Lagrange Multipliers give you candidates for extremes, you have to manually verify whether they actually are maximums or minimums.

[@ar3568row]

@@robinbernardinis yeah right, thank you.

[@Maxwell_Integral]

Great video sir, question. At 11:10 why is it 1,0,-1 but for the second point it’s -3,0,-3 why is that?

[@FranFerioli]

There are two cases: 1) x = - z; when you put this in the other equations you find x = 1 ==> z = -1 2) x = z; when you put this in the other equations you find x = - 3 ==> z = - 3

[@Maxwell_Integral]

@@FranFerioli Thank you so much

[@amrithpurandhar9882]

why can't we take sqrt(x^2+y^2+z^2) it minimum distance rite

[@thomasblok2120]

Is it possible to avoid the Lagrange multipliers altogether by saying that the determinant of all the gradients is zero? This plus however many constraints you have should be enough to make it work as long as you have exactly one less constraint than the number of variables.

[@ysj143]

I'm not a mathematician, so please filter my answer. I solve this problem like what you said. 1st, make an equation that the determinant of all the gradient(g1, g2) is zero. 2nd, dot product of grad f and determint of (grad g1, grad g2) is 0. It can be solved with this method.

[@TheVincent0268]

why do so many video's have a poor sound quality

[@bevanash7683]

I prefer the previous animations as opposed to the handwriting.

[@milanrai6988]

why are you taking such an easy function during tutorial? Can you just take f(x)= x^2y^4z^6 or like so?

[@lgbtthefeministgamer4039]

the complexity from choosing a "harder" function means that the skillset needed to solve it goes deeper into things not directly related to Lagrange multipliers. it's better to introduce new information by focusing specifically on the new information