Results 1 to 7 of 7

Math Help - Calculus 2

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    London
    Posts
    6

    Calculus 2

    f(x,y) = x2 + y2 subject to the constraint xy = 1, find all extrema, if they exist. Where x is in [-10, 10]
    The only (x,y) that satisfy the constraint are =+_1 and y=+_1. In both cases f(x) =2. Here is where I stuck, which one is the max/min?
    Last edited by Aschu; December 29th 2012 at 02:27 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,396
    Thanks
    1847

    Re: Calculus 2

    Do you not understand the meaning of the words "maximum" and "minimum"? If they both have the same value for f, then you can't have one the maximum and the other a minimum. Although it is quite possible that neither is a maximum or minimum.

    There is a theorem saying that a continuous function will achieve both max and min on a compact set (closed and bounded interval will work). Further there is a theorem saying that they must lie either inside the interval, where the derivative is 0, or on the boundary. So it is possible that (1, 1) and (-1, -1) both give a maximum value while the minimum occurs at x= -10 or 10. Or that both (1, 1) and (-1, -1) both give a minimum value while the maximum occurs at x= -10 or 10. Or that the maximum occurs at either x= -10 or x= 10 or the other way around. To tell, evaluate f at those points and see which is the largest and which is the smallest!
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,163
    Thanks
    761

    Re: Calculus 2

    Hey Aschu.

    What is the Hessian matrix evaluated at those points and its determinant?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Nov 2011
    Posts
    12
    Thanks
    1

    Re: Calculus 2

    it's not true that the only x and y that satisfy the constraint are x=y +_1 .... for example the end points x=10 y=0.1 or x=-10 y=0.1 which give f(x,y)= 10^2 +0.1^2 that's higher than 2.

    This is a multivariate question i believe not calculus 2
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2012
    From
    London
    Posts
    6

    Re: Calculus 2

    Thank you very much
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Nov 2012
    From
    London
    Posts
    6

    Re: Calculus 2

    Quote Originally Posted by Ahasueros View Post
    it's not true that the only x and y that satisfy the constraint are x=y +_1 .... for example the end points x=10 y=0.1 or x=-10 y=0.1 which give f(x,y)= 10^2 +0.1^2 that's higher than 2.

    This is a multivariate question i believe not calculus 2
    But after applying Lagrange Multiplier, I arrived to x=y, and the constraint xy=1. Which contradict to ur counter examples.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Nov 2011
    Posts
    12
    Thanks
    1

    Re: Calculus 2

    Lagrange multiplier gives you the minimum in this case, but you have to also check the end points, on the interval x from -10 to 10 for example if x were to be from -100 to 100, and one in the case where x=-100 then from the constraint y=1/-100 and if you plug those values into your f(x,y) gives a greater output compared to 2.

    x=10 and y=0.1 is closed under the constraint 10*0.1=1
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: June 25th 2010, 11:41 PM
  2. Replies: 1
    Last Post: February 11th 2010, 08:09 AM
  3. Calculus III But doesn't require Calculus :)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: June 19th 2009, 05:23 PM
  4. Replies: 1
    Last Post: June 23rd 2008, 10:17 AM
  5. Replies: 1
    Last Post: June 7th 2008, 12:47 PM

Search Tags


/mathhelpforum @mathhelpforum