Session 9: Hessian matrix to find Local maxima, Local minima, Saddle point of a function.

In this video, we will see how to check whether, at the critical points, which we get with the help of partial derivatives, the function is taking maximum, minimum value or its a saddle point.
With the help of Hessain matrix and principal minors, we will see that it's very easy to find the nature of function at the critical points.
We will generalize the notion of Hessian and principal minors from 2x2 order to nxn order. Its a very simple extension and easy pattern to remember.

[@607]

3:58 I think this is incorrect, this would make every point a saddle point unless it is a global optimum, as when you take as a neighbourhood the entire space there will be points of which the image is lower and points of which the image is higher. I think that for a saddle point you should look at α and β separately.

[@DrMathaholic]

I should have said that a differentiable function f which has a critical point (a,b) is said to be saddle point if that condition is satisfied..
Thanks for pointing it out..👏👏

[@abhaykhade539]

Heartly congratulations sir ji...🎊🎉
1k subscribe milestone complete before 1 day of ESE

[@DrMathaholic]

Thank you Abhay😊

[@abedulkareemqasim9441]

YOU ARE GENIOUS

[@DrMathaholic]

@abedulkareemqasim9441 thank you for your kind words 😊

[@OpQp-mc3tr]

amazing content sir please keep helping us
and also come back to coep

[@DrMathaholic]

Thank you.. Happy to hear that videos r still helping to you all..
Coming back😐!! It's difficult ☹️ future ka pata nai, anything can happen

[@OpQp-mc3tr]

@@DrMathaholic 🥺

[@mohsinali-10]

Phenomenal explanation. Thank you!

[@DrMathaholic]

Thank you 😊

[@ghezaegoitom4429]

Thank you so much, I enjoyed every minute of your lecture

[@DrMathaholic]

Welcome and happy to hear that 😊

[@kamrulhassan7157]

Thank you sir ❤

[@DrMathaholic]

@@kamrulhassan7157 welcome 😊

[@hesammoradi8962]

dude you saved me

[@DrMathaholic]

Welcome 😀

[@AJ-et3vf]

Awesome video! Thank you!

[@DrMathaholic]

Welcome:)

[@mohammadrezazebardast3995]

Wish the bests for you

[@DrMathaholic]

Thank you 😊

[@siddharthm16]

Explained Well. Thanks.

[@DrMathaholic]

Glad to hear that.. thank you :)

[@madhavigole178]

1) Critical point (-3,3) - local minima 2) Critical point (-1/2,-1,3/2) - local minima

[@DrMathaholic]

Thank you Madhavi for posting the answer.. appreciated!!

[@madhavigole178]

Sir, is saddle point same as point of inflection?

[@NasrinSultana-jx4lm]

Thank you sir.

[@DrMathaholic]

Welcome 🙂🙂

[@malharpujari1144]

Thank you so much sir ☺️

[@DrMathaholic]

Welcome:)

[@makeitsimple2306]

Thank you from iraq

[@DrMathaholic]

Thank you for your wishes and welcome 😊

[@CHRISKASSMUKEN]

Thank you.

[@DrMathaholic]

Welcome 😀

[@shreeniketfatangare668]

Thank you so much Sir!! All the videos were very helpful ! When will we be seeing the video on Langrange's multipliers?

[@DrMathaholic]

Welcome.
Glad to hear that..

[@DrMathaholic]

About Lagrange multipliers,its late now.. Nobody asked me before and now I am on Diwali vacation till 17th.

[@vishnupantpotdar7610]

Really nice sir ji

[@DrMathaholic]

Thank you Vishnu.😀
Hope you r doing well 😀

[@nagmaniprasad203]

Sir, kindly make an video on quadratic forms

[@DrMathaholic]

Sure.. I will try soon..

[@abhirajrathod5677]

Sir, please can you upload a video on Langrange's multipliers?

[@DrMathaholic]

Its bit late to ask dude.
I am at my home for Diwali..
So next video, may be after 18th..

[@francoisguyvignon]

I was looking for a proof for this - maybe just the 2D case for starters. Anyone has a reference?

[@DrMathaholic]

See book by Thomas calculus

[@ailebao9176]

Sir, if delta1 is zero, what we get? Can you explain for me? Thank you.

[@DrMathaholic]

Then the test fails we cannot conclude..

[@user-iv2yc9yt7j]

Very good explanation.
One comment: while finding the Hessian matrix of the example I think the hand was directed to incorrect matrix term. Thank you

[@DrMathaholic]

Thank you..
Oh okay..
Thanks for pointing it out..

[@cocomatef]

where you got the x^4 from??

[@DrMathaholic]

At what timestamp ??

[@simranmanghwani3514]

If delta1>0, delta2>0 and delta3

[@DrMathaholic]

Then u cannot conclude. Test fails.

[@simranmanghwani3514]

Thank you,sir!!

[@omgawande1567]

1. Critical point (1,1) ,local minima
2. Critical point (-1/2,-1,3/2) , local minima

[@DrMathaholic]

Thank you Om for posting the answer.

[@prabalsarkar3210]

When delta 1 is 0 then which process we go through.. please upload sir

[@DrMathaholic]

Then the test fails..

[@TheBeast-gu9td]

f(x,y) = 2x^3 + 2y^3 - 9x^2 + 3y^2 + 12y
When I calculated fx=0 and fy=0 separately for critical points I got x=0 and x=3,y=1 and y=-2 so should I consider (0,1),(0,3),(3,1),(3,-2)?

[@DrMathaholic]

For safer side, check whether these 4 (x,y) points satisfy fx and fy or not..

[@TheBeast-gu9td]

@@DrMathaholic sir but fx is entirely in terms of x and fy is entirely in terms of y so I could not solve simultaneously both the equations

[@DrMathaholic]

@@TheBeast-gu9td okay..so in that case, u answer is correct...
All 4 points will come

[@TheBeast-gu9td]

@@DrMathaholic thanks sir

[@komalkamaram733]

Sir, could you post the right answers please?

[@DrMathaholic]

I have not solved the questions... 😀
Your answers are not matching with those in comment section?