Show that the product $(x+a)(y+b)(z+c)$, where $a,b,c$ are positive constants, subject to the condition that $xyz = d^3$, where $d$ is a positive constant, has its minimum when $x=$$ad\over\mu$, $y=$$bd\over\mu$,$z=$$cd\over\mu$ and $\mu = (abc)^{1\over3}$.
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Seeing your other question on Lagrange multipliers, it suggests to me that you are trying to learn that technique. If so, please state it, and someone will quickly supply a Lagrange multipliers solution. – Calvin Lin Jul 23 '13 at 15:46
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Hint: Since $a, b, c$ are positive constants, we want to minimize
$$( \frac{x}{a} + 1 ) ( \frac{y}{b} + 1) ( \frac { z}{c} + 1 ) $$
Your conditions are then equal to $\frac{x}{a} = \frac{y}{b} = \frac{z}{c} = \sqrt[3]{\frac{xyz}{abc} }$.
A quick way of proving that $ ( p+1)(q+1)(r+1) \geq (\sqrt[3]{pqr} + 1)^3$ is to apply Jensens on $\ln (e^x+1)$, whose second derivative is always positive.
Calvin Lin
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