Warm up to the second partial derivative test

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An example of looking for local minima in a multivariable function by finding where tangent planes are flat, along with some of the intuitions that will underly the second partial derivative test.

[@theflaggeddragon9472]

Optimization is very close to my heart as a math topic. I proved the second derivative test for functions of n variables using the eigenvalues of the hessian, and so far, it has been the best experience of my mathematical life. I'm 17, looking forward to wayyy more!

[@kamvc72]

so now u r 23 , what majors u hav done?

[@lordcasper3357]

what

[@thomkinrade7116]

I love this guy.

[@LucGendrot]

87 videos in, and his handwriting has REALLY improved with the tablet.

[@PraneshL-yw3lm]

Instead of saying second derivative test , I would prefer saying , have a look at the LAPLACIAN of it , it may be more inituitive , Great video!!!

[@MayankGoel447]

I love how to teach by giving the intuition and then building upon that intuition to derive the formulas. It really helps in remembering and understanding of concepts.

[@bishopseol6992]

This guy is awesome, in uni, they don't have as much graphical explanation as this one does, I am trying to use my time efficiently; this short 11 mins are enough to make you understand how theorem of stationary points actually work

[@anasslyan3490]

Thanks a million. couldn't ask for more explanation!!!

[@lordcasper3357]

the thing i was wondering is mentioned at the end of the video. i was curious about it and when i asked it about my lecturer i couldnt get an answer. LOVE YOU Khan Academy and 3b1b!

[@PhotonDynamics]

Introduction to Maxima and Minima in Single Variable Calculus: 00:00
Determining Nature of Critical Points: 00:11
Extension to Multi-Variable Calculus: 00:59
Example in Multi-Variable Calculus: 01:24
Analyzing the Critical Points: 02:39
Importance of Mixed Partial Derivatives: 03:55

[@cziegl3r]

I'm taking multivar over summer and physics I. Thank you for the vids.

[@sankaracharyadutta9408]

Hey Grant !! Regarding your statement of Saddle points is a new concept in Multi -variable calc and the example about the single variable calculus ---
In single Variable Calculus there exists a point called the POINT OF INFLECTION where the tangent has zero slope but it is neither a maxima or a minima.. INFLECTION points are similar to saddle points because in such points the neighborhoods have different tendencies just like the fact that the partial derivatives have different tendencies here. So
please Refer.
But by the way you are doing a fantastic job by making the viewers
really understand the topics through real 3d graphs... THANKS!! lots of
love...

[@burnttoast5727]

Sankaracharya Dutta thank you, I was losing my mind.

[@burnttoast5727]

Sankaracharya Dutta also, a true saddle point in single variable calculus would be a point where f'(x) and f''(x) are both zero, not just any point of inflection.

[@EmapMe]

Why is f''(x) 0 for a saddle point in single variable calc?

[@MayankGoel447]

​@@EmapMe If x=a is a maxima for f(x) then the function must decrease from thereafter. Alternatively for x>a, f(x)0 then the function will increase. Since f'(a)=0 and f'(x) a this means f'(x) is also decreasing. Hence f''(x) must be negative or mathematically f''(x)0 or f'(x)

[@BudskiiHD]

Thank you!! Great teacher

[@rosawang5965]

Thank you for your videos!

[@thadimmink4128]

I love you. Bless your math soul.

[@shubhambhavsar6933]

a video with zero dislikes. not surprising when Grant Sanderson sir is teaching.

[@adonaythegreat8426]

I like the terms positive and negative concavities

[@armstrongntambale5348]

great staff

[@TheDrB0B]

I would love to see an example where fxx and fyy aren't enough to determine what the critical point is, and you need fxy (10:38)

[@adityakhedekar9669]

watch the next video in series

[@kaymogaladi6878]

What is the name of simulation software that khan academy is using to graph these functions, and where/how can I get it?

[@thefirstprinciplesguy4371]

project manim, I guess.. it's an open source project developed by 3b1b team.. you can use it

[@suraj683]

Why complex roots are not used in finding maxima or minima

[@plaustrarius]

3Blue1Brown! =D what a surprise haha

[@justdoit2521]

How come the local minima correspond to zero on the Y axis when they don't seem to do so when you look at the graph?

[@Algotify]

Did you figure it out?
edit: Nvm, the Y axis is the one left to right. So they are on y = 0. The up down axis is the Z axis

[@theengineered1949]

404 likes, no dislikes... I’ll change that
Now it’s 405 likes

[@nathanaelmasson3445]

you're a fkn legend

[@RiaziMohandesi]

This famous surface is a known as BUTT

[@dinem4624]

I'm sorry but your 2 is really confusing/frustrating it looks like a Z.