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Minimum Value of a Product

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January 18, 2009
inequalitiescalculussymmetryfunctionalgebradomain

Problem Statement

Let x x,y y, and z z be three positive real numbers whose sum is 1 1. If no one of these numbers is more than twice any other, then the minimum possible value of the product xyz xyz is <spanclass=′latex−bold′>(A)</span> 132<spanclass=′latex−bold′>(B)</span> 136<spanclass=′latex−bold′>(C)</span> 4125<spanclass=′latex−bold′>(D)</span> 1127<spanclass=′latex−bold′>(E)</span> none of these <span class='latex-bold'>(A)</span>\ \frac{1}{32}\qquad <span class='latex-bold'>(B)</span>\ \frac{1}{36}\qquad <span class='latex-bold'>(C)</span>\ \frac{4}{125}\qquad <span class='latex-bold'>(D)</span>\ \frac{1}{127}\qquad <span class='latex-bold'>(E)</span>\ \text{none of these}
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