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Homework Help: Max and Min Problem

  1. Apr 24, 2005 #1
    Hello Everyone,

    I am having trouble with a maximum problem and I'm not quite sure where I am going wrong so I will type of the problem and what I have done so far.

    Find two positive numbers whose sum is 18 and the product of the first number and the square of the other is a maximum.

    Here is what I've done so far:

    x + y = 18 ---> y = 18 - x


    P = x(18-x)^2
    P = x(324 - 36x + x^2)
    P= x^3 - 36x^2 + 324X

    To find where there is a maximum I found the first derivative of the equation above:

    dP/dx = 3x^2 - 72x + 324

    This is where I'm stuck, I know I want to make the first derivative equal to zero so I can find the values for the maximum, and verify my answer using the second derivative, but the first derivative cannot be factored. I must be doing something terribly wrong. The first time I did it I got x = 18 and y = 0 which cannot be right, to get that answer I took out a common x value in the equation P equation, but when I did the second derivative test it showed that the answer was actually a minimum. If anyone could give me some direction here I would really appreciate it.
  2. jcsd
  3. Apr 24, 2005 #2
    It doesnt need to be factorable, worst come to worst, you can use the quadratic formula.

    Also dP/dx does simplify.

    [tex] (18-x)^2 - 2x(18-x) = (18-x)(18-x-2x) = (18-x)(18-3x)[/tex]
  4. Apr 24, 2005 #3
    Ok turns out I'm mighty dumb...lol.

    so dP/dx= (x-18)(x-6) = 0
    therefore x can equal 18 or 6
    when you take the second derivative the only number that equals a maximum is 6, so the two numbers are x = 6 and y = 12.

    I think that is right now.
  5. Apr 24, 2005 #4
    This doesnt equal [itex] 3x^2 - 72x + 324[/itex]
  6. Apr 25, 2005 #5
    Actually it does, since 3, 72, and 324 are all divisible by three the polynomial can be simplified to x^2 -24x + 108 which can be factored to be (x -18)(x -6)
  7. Apr 25, 2005 #6
    But you still have to tag a 3 on there.. expand your factors.. you get x^2-24x+108, which is 3 times less than the original polynomial.

    When you solve for x you can eliminate it though, which is what I think ur doing.
  8. Apr 29, 2005 #7
    Oh right, my bad. Just the sort of thing I'd get on a students case about writing, I'm such a hypocrite. :(
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