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Derivative of f(x) to find its maximum and minimum values

  1. Nov 17, 2011 #1


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    Gold Member

    The problem statement, all variables and given/known data
    If λ is a positive constant, determine the maximum and minimum values of
    f(x) = 9(4-3x^2)(λ-λ^-1-x)
    and show that the difference between them is 4(λ+λ^-1)^3. Find the least value of this difference as the parameter λ is varied.

    The attempt at a solution
    I expanded the right-hand side and then did a first d.w.r.t.x

    dy/dx = 81x^2 + 54x/λ - 54λx -36

    I have to equate this to zero, to find either the minimum or maximum value:

    81x^2 + 54x/λ - 54λx -36 = 0

    But i don't know how to proceed next, since the value of λ is unknown, so i cannot solve for x.
  2. jcsd
  3. Nov 17, 2011 #2


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    Science Advisor

    You can certainly use the quadratic formula to solve for x. Of course those values, and the minimum and maximum values of the function, will depend upon [itex]\lamba[/itex].
  4. Nov 17, 2011 #3


    Staff: Mentor

    Mod note - moved from Precalc section.
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