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Math Help - partial differentials

  1. #1
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    partial differentials

    Hey folks,

    I'm working on this problem from the partial diff section of my book:

    Verify that the sum of three numbers, x, y, and z, whose product is constant, is a minimum when they are all equal.

    Just need a hint, I hope, as what I've tried, I'm not sure makes sense.

    I figure I need an equation that somehow includes the constant product restriction in the sum. Maybe something like:

    u = x + y + z
    k = xyz
    and then,
    u = \frac{k}{yz} + \frac{k}{xz} + \frac{k}{xy}

    Then find the total differential, and so on. I know when finding a min/max, each partial has to equal zero. But this one's getting me a little confused.

    Scott
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  2. #2
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    Quote Originally Posted by ScottO View Post
    u = x + y + z
    k = xyz
    and then,
    u = \frac{k}{yz} + \frac{k}{xz} + \frac{k}{xy}
    Notice that,
    x+y+z \geq 3\sqrt[3]{xyz} =3\sqrt[3]{k}
    So that is the minimum value.
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