Find k with $\frac{1}{(3a-b)^2}+\frac{1}{(3b-c)^2}+\frac{1}{(3c-a)^2} \ge \frac{k}{a^2+b^2+c^2}$
Find k $\frac{1}{(3a-b)^2}+\frac{1}{(3b-c)^2}+\frac{1}{(3c-a)^2} \ge \frac{k}{a^2+b^2+c^2}$
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inequality
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0@Mark Bennet find all k, not just one – 2012-11-11
1 Answers
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In other words, you want to minimize $f(a,b,c) = (a^2 + b^2 + c^2)\left(\frac{1}{(3a-b)^2} + \frac{1}{(3b-c)^2} + \frac{1}{(3c-a)^2}\right)$ (presumably for $a,b,c$ real numbers). By homogeneity, we may restrict to the sphere $a^2 + b^2 + c^2 = 1$. Using a Lagrange multiplier, we look for critical points of $F(a,b,c,\lambda) = f(a,b,c) + \lambda (a^2 + b^2 + c^2-1)$. The solution is rather unpleasant. It seems the minimum is approximately $0.808645463814078870$.
EDIT: In fact this number turns out to be the real root of the polynomial $2460640\,{z}^{3}-2787824\,{z}^{2}+905866\,z-210681$
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0Without the restriction, you get critical rays rather than a finite number of critical points. – 2012-11-11