Results 1 to 2 of 2

Math Help - find minimum possible value..

  1. #1
    Newbie
    Joined
    Jul 2009
    Posts
    23

    find minimum possible value..

    can anyone help me with this problem?
    find the minimum possible value of x^2 + y^2 given that x,y are real numbers such that

    xy(x^2 - y^2 ) = x^2 + y^2 , x is not equal to 0.

    thanx
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Hello nh149
    Quote Originally Posted by nh149 View Post
    can anyone help me with this problem?
    find the minimum possible value of x^2 + y^2 given that x,y are real numbers such that

    xy(x^2 - y^2 ) = x^2 + y^2 , x is not equal to 0.

    thanx
    I don't know whether this works, but have you tried the substitution y = xz, since the expressions are homogeneous?

    Then x^2+y^2 = x^2(1+z^2)

    and xy(x^2-y^2) = x^2 + y^2 becomes x^4z(1-z^2) = x^2(1+z^2)

    \Rightarrow x^2 = \frac{1+z^2}{z(1-z^2)}, \, x\ne 0

    So we need the minimum value of x^2(1+z^2) i.e. \frac{(1+z^2)^2}{z(1-z^2)}

    Sorry, I've no more time at present to investigate further.

    Grandad

    Edit: added later

    This does indeed give a solution. The value of the expression must be positive, which means z < -1 or 0<z<1. If you differentiate, and put the result equal to zero, you get a quadratic in z^2, which gives values of z in the permissible ranges of

    z = \sqrt{3 - \sqrt8}= \sqrt2 -1

    and z = -\sqrt{3+\sqrt8}= -\sqrt2 - 1

    Substituting either of these values back gives the minimum value of x^2 +y^2 as exactly 4, but there's a lot of manipulation of surds along the way. (I've checked this numerically on a spreadsheet, and am pretty sure this is correct.)

    Grandad
    Last edited by Grandad; August 12th 2009 at 05:22 AM. Reason: Returned later to add further comments
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Find the possible minimum
    Posted in the Algebra Forum
    Replies: 3
    Last Post: June 9th 2009, 12:43 PM
  2. Find minimum value
    Posted in the Calculus Forum
    Replies: 7
    Last Post: May 2nd 2009, 09:40 PM
  3. Find minimum value of...?
    Posted in the Trigonometry Forum
    Replies: 5
    Last Post: August 29th 2008, 01:33 AM
  4. Find the minimum value
    Posted in the Calculus Forum
    Replies: 5
    Last Post: June 16th 2008, 03:55 AM
  5. find the minimum
    Posted in the Calculus Forum
    Replies: 7
    Last Post: May 6th 2007, 07:20 PM

Search Tags


/mathhelpforum @mathhelpforum