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Find the relative extrema

  1. Jun 16, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the relative extrema of the following function f(x) = (a-x)/(x2-a2)
    where a is a constant, a>0

    2. Relevant equations
    Derivative of f(x), zeroes, quadratic formula


    3. The attempt at a solution

    I think I just screwed a small step in there because my answer doesn't work out (it's supposed to be a(1 + sqrt2) and a(1 - sqrt2)

    f'(x) = [(x2+a2)(-1) - (2x)(a-x)]/(x2+a2)2

    0 = -x2 - a2 - 2xa + 2x2

    0 = x2 - 2xa - a2

    Quadratic formula:

    x = [-b +/- sqrt(b2 - 4ac)]/(2a)

    x = {2xa +/- sqrt[(-2xa)2 - 4(x2)(-a2)]}/(2x2)

    x = [2xa +/- sqrt(4x2a2 + 4x2a2)]/(2x2)

    x = [2xa +/- sqrt(8x2a2)]/(2x2)

    x = [2xa +/- 2sqrt(2)xa]/(2x2)

    x = 2xa(1 +/- sqrt2)/(2x2)

    x = a(1 +/- sqrt2)/x

    ): How do I get rid of the x? If you cancel it doesn't the left side become 1?

    Thank you for your help! <3
     
  2. jcsd
  3. Jun 16, 2011 #2

    Dick

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    Science Advisor
    Homework Helper

    It looks like you meant to say f(x)=(a-x)/(x^2+a^2) in the problem statement. So, yes, you want to solve 0=x^2-2ax-a^2. When you use the quadratic formula you put 'a'=1, 'b'=(-2a) and 'c'=(-a^2). I put quotes around the variables in the quadratic formula so as not to confuse them with the a in the problem. Notice none of them have an x in it.
     
  4. Jun 16, 2011 #3
    WOW that made all the difference! Thank you so much (:!!
     
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