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Math Help - find minimum possible value..

  1. #1
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    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
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  2. #2
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    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 06:22 AM. Reason: Returned later to add further comments
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