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Math Help - Very strange question involving hypersurfaces and the AM-GM Inequality

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    Very strange question involving hypersurfaces and the AM-GM Inequality

    Our textbook has no knowledge about a hyper-surface (or the word is mixed into a random paragraph which I will never find), so I could use some help with this one. I have a potential answer for the second half of this question, which boils down to the AM-GM Inequality, but I'm not sure if my answer is what is being asked for (the context of the question is a bit vague to me).

    Find the minimum distance from the origin to a point on the hyper-surface
    x_1x_2...x_n=p^n, p>0
    ---
    I've been told by another source that, since a hyper-surface is a surface of n dimensions, the minimum distance would occur along the line x_1=x_2=...=x_n, which would mean when they all equal p. However, showing this is proving difficult for me.

    Deduce that, for a_i\geq 0,
    \frac{1}{n}\sum_{i=1}^n a_i\geq \sqrt[n]{a_1a_2...a_n}
    with equality occurring only when all of the a's are equal.
    (HINT: look at a_i=x^2_i)
    ---
    For this second half, this is basically the inequality of arithmetic and geometric means (or the AM-GM inequality), for which I could easily find a proof. However, due to how the question is worded, I'm not entirely sure if a proof is what is being asked of us. As such, I could use some clarification.

    Any help on this one would be appreciated.
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    Quote Originally Posted by Runty View Post
    Our textbook has no knowledge about a hyper-surface (or the word is mixed into a random paragraph which I will never find), so I could use some help with this one. I have a potential answer for the second half of this question, which boils down to the AM-GM Inequality, but I'm not sure if my answer is what is being asked for (the context of the question is a bit vague to me).

    Find the minimum distance from the origin to a point on the hyper-surface
    x_1x_2...x_n=p^n, p>0
    ---
    I've been told by another source that, since a hyper-surface is a surface of n dimensions, the minimum distance would occur along the line x_1=x_2=...=x_n, which would mean when they all equal p. However, showing this is proving difficult for me.

    Deduce that, for a_i\geq 0,
    \frac{1}{n}\sum_{i=1}^n a_i\geq \sqrt[n]{a_1a_2...a_n}
    with equality occurring only when all of the a's are equal.
    (HINT: look at a_i=x^2_i)
    ---
    For this second half, this is basically the inequality of arithmetic and geometric means (or the AM-GM inequality), for which I could easily find a proof. However, due to how the question is worded, I'm not entirely sure if a proof is what is being asked of us. As such, I could use some clarification.

    Any help on this one would be appreciated.
    For the first part of the question, you want to minimise d^2 = x_1^2+ x_2^2 + \ldots + x_n^2 subject to the constraint x_1x_2\cdots x_n = p^n. Use the method of Lagrange multipliers. You should find that x_i = \pm p for each i, and the minimum distance is p\sqrt n.

    Thus if \mathbf{x} = (x_1,x_2,\ldots,x_n) is a point on the hypersurface, it follows that the distance from  \mathbf{x} to the origin must be at least p\sqrt n. In other words, x_1^2+ x_2^2 + \ldots + x_n^2 \geqslant np^2 = n(x_1x_2\cdots x_n)^{2/n}. Now use the hint: let a_i = x_i^2 and rearrange the result to give the AM-GM inequality.
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