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Extreme values problem

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data
    find two positive numbers with product of 200 such that the sum of one number and twice the second number is as small as possible.



    2. The attempt at a solution

    my work:
    xy=200 ==> y = 200/x
    x+2y = s (what we need to minimize)
    x+2(200/x) =s
    x+400x^-1 = s
    1-400x^-2 = ds/dx
    (x^2-400)/x^2 = dx/dx
    (x-200)(x+200)/(x^2) = ds/dx
    crit numbers: 0, 200, -200 (not included because the domain is x>0)
     
  2. jcsd
  3. Nov 12, 2011 #2

    micromass

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    Are you sure that [itex]x^2-400=(x-200)(x+200)[/itex]. What is 200*200??
     
  4. Nov 12, 2011 #3
    oh wow >.> what a silly error.
     
  5. Nov 13, 2011 #4

    Ray Vickson

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    Here is a little hint that applies to ANY problem of the form min f(x) = Ax + B/x with A,B>0 (and we want x > 0). At the min, both terms of f are *equal*, so Ax = B/x. That means that x = sqrt(B/A). (Remembering equality of the two terms is easier than remembering the final formula.)

    By the way, that "equality" result follows from calculus, but can also be obtained without using calculus---that is the basis of so-called "Geometric Programming".

    RGV
     
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