Cauchy Schwarz Inequality | Applications to Problems, and When Equality Occurs
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- Опубліковано 7 січ 2021
- This video is dedicated to applications of the Cauchy Schwarz Inequality, including an application to a problem on the 1995 International Mathematical Olympiad (IMO). We explain why the inequality holds and when it is satisfied with equality. We then show how to maximize a linear function on a sphere using it, and culminate with the IMO problem.
#CauchySchwarz #CauchySchwarzInequality #CauchySchwarzOptimization
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I've been fascinated by inequalities lately, they are powerful!
It's true 😃
This just so inspiring! Hope budding mathematicians are watching! Great video
Nice, Cauchy Schwarz is quite useful in math competitions. The trick is to find a way to properly utilize it as shown in the video.
Agreed 😃
Thanks for the great proof and examples!
Thank you Dr. Mohamed Omar ! 谢谢！
Thank you for once again saving me from overthinking these sorts of problems!
Definitely!
Thank you for the help!
Thanks! I am studying for a University of Maryland math contest and this is really useful!
Fantastic!
Thank you sir. You made this inequality easy to he explained
Definitely
Good day, teacher. I would appreciate it if you explain how we make the transition from exponential numbers to radical numbers.
Classic result; thanks for sharing!
Needed to have it in the mix haha
@@ProfOmarMath And so nicely presented as well -- such a treat, Prof!
Man, You have a great Teaching level, thnks.
Thanks Spyromania!
Sir very nice solution . Thank you .
Definitely Rishav
Amazing !
Thanks!!
In the final step at 9:20, shouldn't the numerator of the lower-bound be (ab+bc+ac)^2?
Extremely helpful sir 🙏
Thanks Atharv!
Again sir a nice one!
Thank you for watching!
Thanks for the video
Definitely!
Thanks sir for the video
Right now l am a seventh grader
I was in IMO math team for iraq this year
Unfortunately iraq is out of the IMO this year
But I will be there for next year and get a medal 🏅
Just saw this. This was wonderful ❤️
@@ProfOmarMath 🌺 thanks sir
Wonderful video
Thanks!
Sir I haven’t understood completely,can u give a prerequisite for this please
I think learning about vectors is the key
Love from india❤
Sir, 4:28 why are you writing y=2x , z=3x as if it is equal ? Please I wanna know. By which condition you hooked it just being parallel to 1,2,3. Please consider if there's some grammatical error. 🙏
Since it was parallel
very important and useful inequality. thanks a lot
Definitely!
In (IMO 1995 ) I also tried to take c=1/ab, then we have ((ab)a/(ab)b+1) +((ab)b/((ab)a+1) +1/((ab)(a+b)) , where a, b>0.
Let (ab)a=k;(ab)b=l, then we calculate minimum for k/(l+1) +l/(k+1) + 1/(k+l) where k,l >0. If I take partial derivative, there is minimum for k=l . Minumum of function f(k)=2k/(k+1) +1/(2k) is for k=1. If k=l=1, then a=b=c=1. So we have thet minimum is 3/2.
I like this analytic approach. I think you get a local min if you check by second derivative test but then an argument is needed to show the local min is a global min
Sir I’m not so good at sigma notation problems , in ur channel is there any videos on that
I would look up "sigma notation" on UA-cam for more!
If you email me I can send you some info on it 😁
@@ProfOmarMath sir can u send the Gmail address , I need a help , today I wrote the math Olympiad and I was not able to do any question , and realised that I was learning math in wrong way plz help me sir 😭🙏
@@chemsonbro7325 Hi. All my info is at www.mohamedomar.org
Check out the videos on my problem solving playlist too!
@@ProfOmarMath I have filled the my details on availability page , by when will I get a mail from u or ur team
Thnqqqq vry much . I want to understsnd it 1 year ago but unable.to coz of my 8th grade teacher says its not for kids. thnqqq
Now it's here!
Quick question at ua-cam.com/video/ZQm8qmQoKos/v-deo.html how we do drop the length of the v vector? Oh wait never mind as I was typing this i realized that the length of the v vector is equal to 1...
then it would be better to delete this post
😁😁. or you could've done that at that time
04:12 Here how do get to the conclusion that vector v and u have to be parallel for x+2y+3z to be sqrt 14? I have completel understanding of vectors and I am learning Calc 3 so feel free to explain using vectors.
Ah yes. The Cauchy Schwarz inequality says the dot product of two vectors is bounded above by the product of their lengths and that equality holds if and only if they are parallel. This happens because the dot product equals the product of their lengths times the cosine of the angle between them, and that is maximized when the angle is 0 or 180 (the latter because we actually take the absolute value of the dot product)
Your notation at the end is a bit sloppy, since it appears that you’re applying the AM-GM mean inequality to (ab + ac + bc) / 2 instead of (ab + ac + bc) / 3 separately to find a minimum of the numerator.
Good vid otherwise 👍
This is the Cauchy-Bunyakovskiy inequality, maybe you are right to call it "-Schwartz", but Bunyakovskiy - this is the name to remember when you talk about that inequality
Does Bunyakovsky's arguement work for a generic inner product space? The wording on wiki page makes sound like Schwartz's argument works in the general case, if true this might explain the current name of the inequality.
@@camerontorrance1992 just the thing to remember from school)
I'll figure it out, I'll reply then