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A plane passes through the point $(a, b, c)$. Find its intercepts with the coordinate axes if the volume of solid bounded by the plane and the coordinates planes is to be a minimum.

What I have tried: Let $\langle x,y,z \rangle$ be the normal vector of the plane. Then, after some calculation, the volume of the solid is$$V=\frac{(ax+by+cz)^{3}}{2xyz}$$

And I found all the first order derivatives, but I can't find the stationary point. How can I do this? Am I using the wrong approach?

Narasimham
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user134927
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1 Answers1

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Assuming $a,b,c > 0$, you may also assume $x,y,z \geq 0$ and $xyz=1$.

Then, by the AM-GM inequality:

$$(ax+by+cz)^3 \geq 27(axbycz) = 27 abc.$$ Equality happens iff $ax=by=cz$, or : $$ (x,y,z) = \frac{1}{(abc)^2}\left(bc,ac,ab\right).$$

Jack D'Aurizio
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