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

  1. Nov 17, 2011 #1

    sharks

    User Avatar
    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

    HallsofIvy

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    Staff Emeritus
    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

    Mark44

    Staff: Mentor

    Mod note - moved from Precalc section.
     
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