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Math Help - Differentiation Applications 2

  1. #1
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    Differentiation Applications 2

    I have two more quick questions:

    The sum of two numbers is 10. What are the two number if the sum of their squares is a minimum?

    What numbers, when squared and added to their own squared reciprocals, lead to a minimum value?

    Help is greatly appreciated.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by holmesb View Post
    I have two more quick questions:

    The sum of two numbers is 10. What are the two number if the sum of their squares is a minimum?
    Let the numbers be x and y. Then
    x + y = 10 \implies y = 10 - x

    We wish to minimize f(x, y) = x^2 + y^2

    Well,
    x ^2 + y^2 = x^2 + (10 - x)^2 = 2x^2 - 20x + 100

    Critical points of this are at:
    \frac{d}{dx}(2x^2 - 20x + 100) = 0

    4x - 20 = 0

    x = 5
    (I leave it to you to prove that this is a minimum.)

    Thus y = 10 - 5 = 5 as well.

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by holmesb View Post

    What numbers, when squared and added to their own squared reciprocals, lead to a minimum value?
    Call the number x. The we want to minimize
    x^2 + \frac{1}{x^2}

    So take the derivative, set it equal to 0 and solve for x.

    -Dan
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