Results 1 to 4 of 4

Thread: Local extremum (2 variables)

  1. #1
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1

    Local extremum (2 variables)

    I have to find the critical points of the function f and to tell whether they are local minimum, local maximum or saddle point.
    The function is $\displaystyle f(x,y)=x^2y^2$.
    By intuition the point (0,0) is a local minimum.
    Formally, I found that $\displaystyle f_x(x,y)=0 \Leftrightarrow 2xy^2=0$, so $\displaystyle x=0$ and $\displaystyle y=$ any value. (I'm not convinced since y is treated as a constant and it might be 0, while in this case x could be any other number different from 0. Hence my "so" is not an implication in fact! I don't understand what I'm doing wrong)

    $\displaystyle f_y(x,y)=0 \Rightarrow y=0$ and $\displaystyle x=$ any value. (Once again I don't trust my implication)
    Hence the critical point is $\displaystyle (0,0)$.

    From my class notes, if $\displaystyle f_{xx}(0,0)f_{yy}(0,0)-f_{xxy}(0,0)=0$, then I cannot conclude about the nature of the critical point. Unfortunately it happens here, and there's no theorem that can help me.
    If it was the case of a transcendantal function, I would have approximate it by a Taylor's polynomial and checking if the polynomial reaches a minimum or a maximum or a saddle point, but the problem is that the function is a polynomial. (by the way I'm not even sure that approximating a function via a Taylor's polynomial assegurate that the approximated function reaches the same critical points. Maybe there's a theorem about it but I never read it yet).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Spec's Avatar
    Joined
    Aug 2007
    Posts
    318
    All the critical points are positive-semidefinite since $\displaystyle Q(h,k)=2y^2h^2,\ y\in R$ or $\displaystyle Q(h,k)=2x^2k^2,\ x\in R$ (depending on the critical point)

    Because of that, you can't use the quadratic form to see if the point is an extremum or not.

    $\displaystyle x^2\geq 0,\ y^2\geq 0 \implies x^2y^2\geq 0$

    $\displaystyle f(x,y)=0 $ for all points $\displaystyle (x,y)=(0,C),\ C\in R$ or $\displaystyle (x,y)=(C,0),\ C\in R$, so these points are local minima ($\displaystyle f'_x=0,\ f'_y=0$ here as well).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Amer's Avatar
    Joined
    May 2009
    From
    Jordan
    Posts
    1,093
    Quote Originally Posted by arbolis View Post


    From my class notes, if $\displaystyle ?!""f_{xx}(0,0)f_{yy}(0,0)-f_{xxy}(0,0)""!?=0$, then I cannot conclude about the nature of the critical point. Unfortunately it happens here, and there's no theorem that can help me.
    .
    The formula not like that but like this

    $\displaystyle D = f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-f^2_{xy}(x_0,y_0)$
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1
    Quote Originally Posted by Amer View Post
    The formula not like that but like this

    $\displaystyle D = f_{xx}(x_0,y_0)f_{yy}(x_0,y_0)-f^2_{xy}(x_0,y_0)$
    Thanks a lot! My professor wrote the $\displaystyle ^2$ but seemed right after the $\displaystyle _x$. She didn't prove anything so I took the formula I saw, without understand why it is like it is. I miss a good book on the subject. (I can't borrow Stewart's one on vector calculus since too many people borrow it)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. local man, local min, inflection point problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 27th 2011, 11:44 PM
  2. Replies: 4
    Last Post: Mar 21st 2011, 01:23 AM
  3. Replies: 6
    Last Post: Jan 5th 2011, 02:34 AM
  4. Local extrema in multiple variables
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Apr 21st 2009, 04:06 AM
  5. extremum
    Posted in the Algebra Forum
    Replies: 9
    Last Post: Aug 1st 2007, 04:31 AM

Search Tags


/mathhelpforum @mathhelpforum