Classify the Critical Points (Advanced)

[LiveLowGrow]

Good Day,

My professor historically has put more advanced questions on the final exams than can be found in our textbook. My supplements and extensive search engine use have not allowed me to get any further...Ive hunted through as many examples as I could find but cannot seem to find a guiding light when it comes to this huge polynomial.. this is as far as I've gotten and my exam is in 18 hours... any help would be appreciated.


Classify the critical points of f(x,y)=x^3+y^3+x^2+y^2-2x-3y


I get my first order derivatives of Fx=3x^2-2 and Fy=3y^2+2y-3

..then I complete the squares to get roots of x=(sqrt7)/3 - 1/3 and x= - (sqrt7)/3 - 1/3

similarily I get y=(sqrt10)/3 - 1/3 and y= - (sqrt10)/3 - 1/3


D=FxxFyy-(Fxy)^2 gives me

(6x+2)(6y+2)-0
=36xy+12x+12y+4

and this is where I get lost...any help would be appreciated...

 

[DryRun]

Classify the critical points of f(x,y)=x^3+y^3+x^2+y^2-2x-3yI get my first order derivatives of Fx=3x^2-2 and Fy=3y^2+2y-3

..then I complete the squares to get roots of x=(sqrt7)/3 - 1/3 and x= - (sqrt7)/3 - 1/3

similarily I get y=(sqrt10)/3 - 1/3 and y= - (sqrt10)/3 - 1/3D=FxxFyy-(Fxy)^2 gives me

(6x+2)(6y+2)-0
=36xy+12x+12y+4

and this is where I get lost...any help would be appreciated...

Use LaTeX to make your mathematical expressions clearer to read and interpret.

$$f(x,y)=x^3+y^3+x^2+y^2-2x-3y$$
$$f_x = 3x^2+2x-2$$ and $$f_y=3y^2+2y-3$$
You made a mistake in finding ##f_x##

 

[LiveLowGrow]

oh thank you...I am new to this sort of thing...and I will follow up on that tutorial as soon as i can...

...the Fx mistake was actually a mistype on my part...the correct Fx is written here on my paper...sorry and thanks for that...I will be more aware in the future...

 

[CAF123]

You have the solutions of ## f_y = 0## incorrect. Once you have those solutions, you will be able to find all critical points. Then evaluate the Hessian at each of your critical points and check the Second derivative test.

 

[LiveLowGrow]

Hey all...thanks for the info ...I checked my fy=0 using the complete the squares method and the quadratic formula and they both give me the same result.. :(

..the problem with the questions I'm doing is that he never taught us the Hessian in class...and its not even in our Stewarts textbook...I first saw it briefly yesterday.. a link to a good walk through would be ideal if such a thing exists

..all of these questions on the previous exams I am studying are much more difficult than what he taught...

...our midterms class averages were 51% and 52% and my final exam is tonight ..
..I have already spent hours solving other weaker critical point and max min questions ...

this question and the other I posted on this forum 2 days ago are the only I cannot solve yet...

this question is worth 16% on 2 of the past 4 exams.. and my other post the question is on all 4 past finals and worth 16% as well...

...thank you so much for your advice and guidance to date but I am going to spend the next 7 hours rehearsing the problems I already have solved ...and if I feel confident Ill hopefully have the time to learn the Hessian method and fiddle around with this more...

...Thanks so much for all the help...take care for now... :D

 

[LiveLowGrow]

Ok I managed to get it...:D I will get my Latex game together for any future post and thanks CAF123 for the step by step instructions and for letting me know about Hessian...

cheers guys..