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Math Help - Writing out the minimum function

  1. #1
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    Writing out the minimum function

    Hi you all

    I have a quick and properly really easy question.
    I saw around the internet that the minimum function can be written out:

    min\left\{ a,b\right\} =\frac{1}{2}\cdot(a+b-|a-b|)

    I haven't ever seen that before, so i am interested in knowing when it works and when it doesn't (and to some extent why it's true). I need to differentiate a minimum function in my exam in microeconomics so it would be a great help!

    Best regards.
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  2. #2
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    Quote Originally Posted by Flemming View Post
    Hi you all

    I have a quick and properly really easy question.
    I saw around the internet that the minimum function can be written out:

    min\left\{ a,b\right\} =\frac{1}{2}\cdot(a+b-|a-b|)

    I haven't ever seen that before, so i am interested in knowing when it works and when it doesn't (and to some extent why it's true). I need to differentiate a minimum function in my exam in microeconomics so it would be a great help!

    Best regards.
    Not sure about the differentiation part, but proving it's true is straightforward:

    Case: a = b

    Then \frac{1}{2}\cdot(a+b-|a-b|) = \frac{1}{2}\cdot(a+b) = a = b

    Case: a < b

    Then \frac{1}{2}\cdot(a+b-|a-b|) = \frac{1}{2}\cdot(a+b-(b-a)) = \frac{1}{2}\cdot(2a)=a

    Case: a > b

    Then \frac{1}{2}\cdot(a+b-|a-b|) = \frac{1}{2}\cdot(a+b-(a-b)) = \frac{1}{2}\cdot(2b)=b
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  3. #3
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    Oh, you are right, I should have been able to figure that one out my self. But thank you.

    Does anyone have any comments about the differentiability of the expression?

    Best regards,
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  4. #4
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    if you have

    a(x) = min(b(x),c(x))

    Except in special cases, The function will not normally be differenciable at the point b(x) = c(x). it will be differenciable at other points provided b(x) and c(x) are differenciable.


    What is the economics problem you are trying to solve? It may be easier to assume min(a,b) = a; and check that your solution is at a point a \leq b. If that doesn't work then assume min(a,b) = b and check that your solution is at a point b \leq a.
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  5. #5
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    I needed to differentiate a function with respect to two different variables and find the quotient of them in order to get a function which hopefully could tell me more about the function which i originally was assigned to find.
    Even though it my problem seemed far fetched your hint of dividing into different situations actually worked like a charm. Thank you!
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