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

    Deduce that, for $\displaystyle a_i\geq 0$,
    $\displaystyle \frac{1}{n}\sum_{i=1}^n a_i\geq \sqrt[n]{a_1a_2...a_n}$
    with equality occurring only when all of the $\displaystyle a$'s are equal.
    (HINT: look at $\displaystyle a_i=x^2_i$)
    ---
    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
    $\displaystyle x_1x_2...x_n=p^n$, $\displaystyle p>0$
    ---
    I've been told by another source that, since a hyper-surface is a surface of $\displaystyle n$ dimensions, the minimum distance would occur along the line $\displaystyle x_1=x_2=...=x_n$, which would mean when they all equal $\displaystyle p$. However, showing this is proving difficult for me.

    Deduce that, for $\displaystyle a_i\geq 0$,
    $\displaystyle \frac{1}{n}\sum_{i=1}^n a_i\geq \sqrt[n]{a_1a_2...a_n}$
    with equality occurring only when all of the $\displaystyle a$'s are equal.
    (HINT: look at $\displaystyle a_i=x^2_i$)
    ---
    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 $\displaystyle d^2 = x_1^2+ x_2^2 + \ldots + x_n^2$ subject to the constraint $\displaystyle x_1x_2\cdots x_n = p^n$. Use the method of Lagrange multipliers. You should find that $\displaystyle x_i = \pm p$ for each i, and the minimum distance is $\displaystyle p\sqrt n$.

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