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I am studying for my math final and I just wrote the practice final. Unfortunately there are no solutions and I am completely lost on how to do this problem. If anyone could help I would really appreciate it.

Question: Show that the product of the x, y, and z intercepts of any tangent plane to the surface xyz = 1 in the fi rst octant is a constant.

I tried rearranging the equation to $z=\frac1{xy}$ then I tried to find the tangent plane using the formula $$z=f(a,b)+f_1(a,b)(x-a) + f_2(a,b)(y-b)$$ but I got confused and it ended up being a big mess. Anyways if anyone could lend a hand here I would really appreciate it.

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Searching on Google for product xyz tangent lead me to this file. (On the website or Robert E. Megginson.) –  Martin Sleziak Apr 29 '13 at 8:52

2 Answers 2

Look at question two in this set of examples: http://www.math.ubc.ca/~malabika/teaching/ubc/fall08/math263/problem-set1-solution.pdf

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I find that links by themselves do not make particularly good answers (and we should also be wary of link rot). Could you please edit this answer to add an explanation as to how this is relevant to the question? –  Douglas S. Stones Apr 30 '13 at 13:02

The gradient of $f(x,y,z)=xyz$ at $(a,b,c)$ is $\langle bc,ac,ab\rangle$ which is the same as $\langle 1/a,1/b,1/c \rangle$. Therefore, the tangent plane has equation $$\frac{x-a}{a}+\frac{y-b}{b}+\frac{z-c}{c}=0$$ Rearrange it into $$\frac{x }{a}+\frac{y }{b}+\frac{z }{c}=3$$ and then divide by $3$ to obtain the intercept form of the plane equation:
$$\frac{x }{3a}+\frac{y }{3b}+\frac{z }{3c}=1$$ (The intercepts are what you see in the denominators.)

This generalizes to $\mathbb R^n$: the product of $n$ intercepts of tangent hyperplanes to $x_1\cdots x_n=1$ is $n$.

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