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Math Help - Minimum

  1. #1
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    Minimum

    Find two numbers whose difference is 100 and whose product is a minimum.

    The problem was given without a function and so I'm not really sure what to do.
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  2. #2
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    Quote Originally Posted by bobsanchez View Post
    Find two numbers whose difference is 100 and whose product is a minimum.

    The problem was given without a function and so I'm not really sure what to do.
    a-b=100

    ab=minimum

    a=100+b\ \Rightarrow\ b=a-100

    ab=a(a-100) in terms of "a".

    Differentiate with respect to "a" and equate to zero to find "a" giving minimum product.
    Then obtain "b".
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  3. #3
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    So...

    a= 50

    b= -50
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  4. #4
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    Quote Originally Posted by bobsanchez View Post
    So...

    a= 50

    b= -50
    Yes,
    that's it.
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  5. #5
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    Another problem similar to that one...


    Find two positive numbers whose product is 81 and whose sum is a minimum.


    I've started it with the method you suggested but I don't know where to go from a = 81/b. Is this where I differentiate?
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  6. #6
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    Quote Originally Posted by bobsanchez View Post
    Another problem similar to that one...


    Find two positive numbers whose product is 81 and whose sum is a minimum.

    I've started it with the method you suggested but I don't know where to go from a = 81/b. Is this where I differentiate?
    Yes, you are forming an equation in one variable.
    Differentiate the sum and equate to zero and solve.
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  7. #7
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    I get 0.
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  8. #8
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    Since the problem asked you to find two numbers, I have no idea what you could mean by "I get 0".

    What function are you differentiating? The problem asked for two numbers "whose sum is a minimum". The sum of two numbers, x and y, is f(x,y)= x+ y. That is what you want to differentiate. You are also told that their product is 81: xy= 81 so y= 81/x so that f(x)= x+ 81/x= x+ 81x^{-1}.
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  9. #9
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    Ah, I see what I did. I tried to differentiate y = 81/x.

    Alright, so the answer is x = 9 and y = 9?
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  10. #10
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    Quote Originally Posted by bobsanchez View Post
    Ah, I see what I did. I tried to differentiate y = 81/x.

    Alright, so the answer is x = 9 and y = 9?
    Ok,
    you need to understand the point of having 2 clues.
    One clue allows you to write one variable "in terms of the other".

    Then you can write an equation in one variable only.

    y=81/x is only one component of the sum equation.
    You want the sum to be a minimum, but you must work with the entire sum formula.

    xy=81
    x+y=sum.

    y=81/x

    Therefore x+81/x=sum.

    x+81x^{-1}=\ sum.

    sum is a minimum, so the derivative, wrt x, of the sum is zero.

    See?
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  11. #11
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    Yeah, that makes sense to me. I just have trouble seeing the relation between the two and putting them in a form I can work with to solve the problem.
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