Multi-variable Optimization & the Second Derivative Test

Finding Maximums and Minimums of multi-variable functions works pretty similar to single variable functions. First,find candidates for maximums/minimums by finding critical points. Critical Points are where the partial derivatives with respect to x and y are both zero. Then we classify each critical point using the second derivative test. In the multivariable case, there is a new option beyond max/min/neither, there is also the case of the saddle point. The second derivative test involves computing the Hessian, the determinant of a matrix that helps decide whether points are maximums/minimums/saddle or inconclusive. We sketch the geometric intuition behind the Hessian.

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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.

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[DrTrefor]

Note that the conditions of the 2nd derivative test should also demand the 2nd partials are CONTINUOUS.

[Mobinahere]

This playlist is, without a doubt, FASCINATING! Thank you!

[randompianonoises]

SO THAT'S WHAT THE MIXED PARTIALS ARE FOR THANK YOU SIR

[zeeshanmouzam]

THE BEST bet none of my teachers (in India) can even think of this...

[ossahmadrezaazimikohnabi]

I love that I live in an era where when I get confused studying my text book I can just come yo YouTube and have someone explain the same thing to me in 3D.

Thank Dr, great lecture.

[chyldstudios]

Clear explanation of mixed partial derivatives. More videos on the Hessian matrix and its applications.

[frjrhdi]

The mixed partial derivative fxy=fyx is the concavity on the line x=y, explained in 10:30, is great.

Thank you.

[BrytonWA]

Thank you for the intuitive explanation of the mixed second partial derivative and what it is representing. I took some time sitting and contemplating on my own before I came to a similar conclusion. It's nice to hear some confirmation of that!

[sazdaz]

Exceptionally good and easy to follow explanation! I was having trouble understanding how to interpret the calculation for the 2nd derivative test and this definitely helped a lot.

[noctua]

This is the best explanation I've ever seen. Thank you my man!

[dilekpusat]

Thank you for clear and engaging teaching. It was wonderful.

[jonpritzker]

Dang, I posted two comments here, first one I thought I was so clever because your diagram and explanations were so clear I got the idea before you said the words but then you said exactly the words and the second comment was a question and then you explained that question. You are an excellent teacher

[riyaverma]

Thank you so much! Your videos are helping me survive multivariable calculus :)

[isakwatz]

thanks man! never knew what mixed partials are supposed to mean but you made it crystal clear! :D

[badrmoubile]

For the people who don't understand the pink part, it is just the result of the Hessian 2x2 matrix. that you solve by doing, a.d-b.c

[thinhvucuong]

Great video!! I really looking forward to your new video in this topic. Thank you!

[aashsyed]

so that is what it has application! thanks! you explain like 100x better my teacher.

[hareshwedanayake]

Thank you so much! You are such a great teacher. This simplified this topic so much for me

[GoutamDAS-lswb]

Thanks so much Professor Bazett for this excellent, lively presentation! You convey your enthusiasm to your audience. They are helping me so much as I review multivariable calculus being offered on edX. I am very, very grateful.

[joaomattos]

Fantastic way of teaching!!! I recommend the classes here in Brazil!