Local Extrema, Critical Points, & Saddle Points of Multivariable Functions - Calculus 3
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- เผยแพร่เมื่อ 28 ก.ย. 2024
- This calculus 3 video explains how to find local extreme values such as local maxima and local minima as well as how to identify any critical points and saddle points in a multivariable function such as f(x,y).
Lines & Planes - Intersection:
• How To Find The Point ...
Angle Between Two Planes:
• How To Find The Angle ...
Distance Between Point and Plane:
• How To Find The Distan...
Chain Rule - Partial Derivatives:
• Chain Rule With Partia...
Implicit Partial Differentiation:
• Implicit Differentiati...
________________________________
Directional Derivatives:
• How To Find The Direct...
Limits of Multivariable Functions:
• Limits of Multivariabl...
Double Integrals:
• Double Integrals
Local Extrema & Critical Points:
• Local Extrema, Critica...
Absolute Extrema - Max & Min:
• Absolute Maximum and M...
________________________________
Lagrange Multipliers:
• Lagrange Multipliers
Triple Integrals:
• Triple Integrals - Cal...
2nd Order - Differential Equations:
• Second Order Linear Di...
Undetermined Coefficients:
• Method of Undetermined...
Variation of Parameters:
• Variation of Parameter...
________________________________
Final Exams and Video Playlists:
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Fxx=-6 ; Therefore it is local maximum as Fxx
Yes I agree
yes
Ya right
Agreed
After seeing that, I went straight to de comments😂
For problem one it should be local max. D < 0 and fxx(a,b) < 0 , then f(a,b) is a local maximum.
D > 0*
Yeah, I noticed that too
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Shouldn't (2,2) be a local maximum? Since fxx is negative?
are the other examples correct? I dont want to study incorrect material....
@@pjk7685 I believe so.
Yeah thats correct
@@pjk7685 Yes, the other examples are correct. Notice how he contradicted himself later saying that fxxfyy - fxy > 0 meaning that it's a local min (which it is).
Only the first example is incorrect.
@@pjk7685 He may make a silly mistake while solving as any other human but he never teaches anything wrong and never will.
Aight imma watch this whole channel before finals lol
Just to remind you...
Yeah, same tbh. Better than lectures
the point (2,2) should be a local maximum because fxx is negative and D is positive!
were are u from?
what does the letter D mean plz help???
@@imranahmed-mt8wg D --> second partial derivative test
@@imranahmed-mt8wg Discriminant or Hessian of function f
true
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On the second example, why do we chose (0,0) , (1,1) and (-1,-1) as our points of interest?
These points satisfy the two equation
If that is so, we can have points like (8,2) , (27,3).... also right? Do we need to consider points like these?
@@sadhiyausama9504 Those points aren't valid, as the
fx value for f(8,2) would be 8(8^3-2)= 8(510) =/= 0
Only the fy value would be 0, 8(2^3-8) = 8(8-8) = 0
Only points that make both the derivatives 0 work
The reason why f(-1,-1) works is because
fx = 8(-1^3-(-1))= 8(-1+1) = 0
fy = 8(-1^3-(-1))= 8(-1+1) = 0
@@trollguyawsomeness oh yeah!! Correct..I didn't think about fx..Thank you so much
The method isn't shown, but you just solve for a variable in the 1st equation and substitute for it in the 2nd so you only have an equation with respect to a single variable. Like this:
First, our initial equations are:
[1] 8(x^3 - y) = 0, [2] 8(y^3 - x) = 0
We can divide equation [1] and [2] by '8' on both sides, as they equal 0. This will simplify stuff.
[1] x^3 - y = 0, [2] y^3 - x = 0
Let's try solving [1] for y:
-> x^3 - y = 0 x^3 = y
Well that was easy, wasn't it? Now we substitute for 'y' in equation [2] with the new value.
[2] (x^3)^3 - x = 0 x^9 - x = 0 < [Factor] > x(x^8 - 1) = 0
Now to solve this, we have 2 solutions:
(1) x = 0, (2) x^8 - 1 = 0
Since the exponent on 'x' in (2) is even, both x = 1 and x = -1 are solutions.
Now finally to get their respective 'y', just plug it back into our original substitution.
(1) x = 0, y = 0^3 = 0
(2) x = 1, y = 1^3 = 1 OR x = -1, y = (-1)^3 = -1
So our points of interest become: (0, 0), (1, 1), (-1, -1), as they are the only ones which satisfy both equations. They are the only possible critical points for the function. To figure out whether they are relative/local extrema, or simply saddle points, we use the double derivative test!
If you could do triple integrals with finding the bounds, youll be a life saver
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the organic chemistry teacher is an angel
6:07
Why did we say this point is local min , and the rule says if the second derivative is negative , it will be local max ?
+
Thank's so much on your effort.❤❤
Caught that as well, I assume it was a blunder
yeah same, i changed my notes and then realized oh wait no... he is wrong
@@zeitgeistsandmicrocosms blunder :DDDDDDDDDDd
hey friend, love your work always and forever but please check your solution to the first question. Should be local max.
I learned more from you than from the lectures of my professor.
Could you do a video on sketching double and triple integrals?
Can you suggest me
@The Organic Chemistry Tutor, I think the example number 1, the point is a local maximum.
the first math should be local maximum ,thnx,great video,huge fan,just doing my work,so that others can follow through
Local min & max for multivariable functions
Great and helpful video. Thanks. Keep the good work up.
Why are 00 11 -1-1 points of interest? Are they given or is it just what you use when dealing with a cubic equation?
We set f(x) = 0 and f(y) = 0, then we take out the two values that gives us f(x) = 0 and f(y) = 0 then use them as Points for D
really helped for my assignment thanks a bunch
luckily his old videos will help me
what is D????
May God bless your generation 🙌
My second lecture...first is my mother 😊
İn my opinion you are wrong in first example. Because there are should local max(Atəşə salam)
what if d is greater than 0 but fxx is zero ? as in (0,0)
Hi, can anyone please post a link on how to factor multivariable functions as the one used in the video in form of binomials e.g. (x-y)(x+1)(x-1)
Calc 3 looks fun. Yum.
thankyou!!! you saved me
Hello your first question you have made a mistake in determining the point (2,2) you said local minima and according to your notice at the beginning of the video it was local maxima as D was positive and fxx was negative so please correct
BRO YOUR COMING IN FUCKING CLUTCH FOR MY CALC III EXAMS
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how come the first question had points of interest but the second question had points of interest? Are the points of interest a given or a general implication or something?
Respected sir in the begining of vedio you said if D>0and and f
it a mistake the correct answer is local maximum.
@@ntandomasango3563 okyyy
Once again, thanks !
point 2,2 is a LOCAL MAX.
thank you
Thank you soo much!
i didnt understand the point of interest, can anyone explain please?
Thanks ❤
9:05 i could do x=2 y=8 and the equation would be true whats up with that?
Tackar som fan my friend
Bless your soul
Thank you sir
note to self:
correction in first qn: local maxima
This is calculus 3? What? I just took an exam for it for calculus 1.
Shfjj Snndjd , you learn first derivative test and second derivative test on calculus one. The difference now is you have a multi variable function, since you learn partial derivatives on calculus 2.
Why don't you just refer to D as the hessian, the determinant if the Hessian matrix? Also, if you use eigenvalues of the Hessian, two positive eigenvalues suggest a minimum, two negative give a maximum, and two mixed signs give a saddle point.
how did he get those point of interests for the second question. Did he just randomly choose them?
www.google.com/gasearch?q=find%20all%20points%20where%20the%20function%20has%20any%20relative%20extrema.%20identify%20any%20saddle%20points%20x%5E3-12xy%2B8y%5E3&tbm=&source=sh/x/gs/m2/5#fpstate=ive&vld=cid:fbcfc846,vid:UID893EosM8,st:0
He breaks it down in this video
what if you have variables in your D value
How do you find fxy????
I SPEND 5 HOURS ON THE GODDAMN CAlCULUS BOOK AND IT DIDN’T MAKE SENSE UNTILL I SAW YOUR VIDEO THAT ONLY TOOK 14 MINUITS
Thanks ❤🌹🙏
what is the proper name for 'D'? I don't know how else to search for it
late but D stands for discriminant
What if d=0
What if D=0??
If D=0, then no conclusion can be drawn
Why fxy is 0? Isn't fxy= -6×(-4)?
I can entrust my midterm on him without a second thought
Thank you Mark Wahlberg
lifesaver🙂
what fxx is 0?
roland Banda The test is considered inconclusive I think
How to calculate fxy?? Why equals to 0??
there is no variable that contains xy. If you derive f(x,y) in respect to x, all the y is considered as a constant. Then you derive in respect to y again, all the x are gone, which is 0
@@nostestwu8689 You don't derive. You differentiate. Differentiating is the same as taking the derivative, but deriving has a completely different meaning than taking the derivative.
it shows you have no teaching training... you skip over some details that if you don't know you lose track of the whole thing
Some people excel at finding the x's and y's and derivatives and blah blah blah. Still I ask, what the fukk is this?? It looks insanely useless.
the future of society is dependant on this man
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@@aswininaidu same here. He's been there for me since my AS and A'level days.
How do you find the points of interest though? When your partials have both x and y in them, I mean
Real
how did he find those three points at the last problem (14:34) (0,0) (1,1)(-1,1) . I know that he said plug in 0 ,1,-1 but I'm not sure why he did that.And where did he get those values to plug in
How is this youtube channel better than my school wtf am i doin with my life?
on the second question, how did you determine that those were the points of interest?
this
You first solve the two first partial derivative equations then you will find critical points from there
@@khiziiikhan how
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P(2,2) is local max not min, u contradicted yourself.
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3 hours before my final🤧🤧
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same here
Sounds like I wrote it myself
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You made a mistake fxx0 it's not local min but local max
im not gonna lie, math been having me fight for my life in the arena for the last couple weeks. thank u so much for this video. gonna secure atleast B because of your help. u a real one.
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Pls how do you get the numbers you plug in as 2 for the first question?
literally more useful then my calc III professor
why did you choose the points (0,0) (1,1) and (-1,-1) for the second problem?
did you know why??
Did u find out y?
How did he decide the point of interest as (0,0) & (1,1) in Q2?
those were all the (x,y) pairs for which the equation was equal to 0 i.e fx = 0, fy = 0
This guy has been carrying my education for the last 3 years
I think you confused on the example - when D>0 and f(xx)
shouldn't it be local max ?
the point (2 , 2) is local max not min!!
Professor Organic Chemistry Tutor, this is another fantastic video/lecture on Local Extrema, Critical and Saddle Points in Calculus Three/Multivariable Calculus. The problems associated with this topic are messy/problematic from start to finish. Once again, thanks to the great viewers for finding and correcting the error at the 6:04-minute mark in the video.
there is a mistake, we have local max , not min
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scrolling through the comment box saved my 2 minutes of searching inside the textbook for the mistake. :P
Could you make a video about you
I had some fun trying to graph this multivariable function from 7:05 and got this on Geogebra: 'imgur a 48VnSO7' (youtube comments aren't so nice with links, put a 'forward slash' between the 'com' and 'a' and the value. Remember that 'com' goes after imgur.
I noticed that the local minimum points reach '8', as calculated for both f(1,1) and f(-1, -1), but how come the graph intersects the z-axis at '12' instead of the '-64' we calculated through f(0, 0)?
What happens when fxx equals zero?
It's inconclusive 💀
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The first problem in which you derived a critical pt of (2,2) represents a local max, not min.
Anyway cheers!