Local Extrema, Critical Points, & Saddle Points of Multivariable Functions - Calculus 3
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- Опубликовано: 5 ноя 2019
- 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...
________________________________
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Fxx=-6 ; Therefore it is local maximum as Fxx
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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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How do you find the points of interest though? When your partials have both x and y in them, I mean
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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.
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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.
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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
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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
Aight imma watch this whole channel before finals lol
Just to remind you...
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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
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2:25 yo that's actually helpfull thx, now I can imagine what's the local min, local max, saddle point! thank you again :)
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@The Organic Chemistry Tutor, I think the example number 1, the point is a local maximum.
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Could you do a video on sketching double and triple integrals?
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hey friend, love your work always and forever but please check your solution to the first question. Should be local max.
thank you
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)
Thank you sir
the first math should be local maximum ,thnx,great video,huge fan,just doing my work,so that others can follow through
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!
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Thanks
is it possible for the local min or max be equal to zero, if so, whats the conclusion?
thx a lot
Pls how do you get the numbers you plug in as 2 for the first question?
Thanks ❤🌹🙏
In my Calc book it says the hessian (D in this case) = B^2 - AC but here I see it calculated as AC - B^2
Thaks
Pleased i need the playlist of this video
D > 0 and Fxx < 0 => Local Max
Thank u very much sir... God bless you
Sir i will need videos on differentiatial equation and partial differentiation from first principle of multivariable.. can u help please
How do you determine the points of intrest
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Absolute life savior, god damn
Local min & max for multivariable functions
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 ?
+
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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
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how do we calculate fxy please
P(2,2) is local max not min, u contradicted yourself.
How was the points of interests for the 2nd question found???
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i just want to see his earlier vids
How can i implent this when i have 3 variables and not only two ?
I think you confused on the example - when D>0 and f(xx)
You made a mistake fxx0 it's not local min but local max
what happens in f(x,y)=(e^x)cosy?