Tìm GTNN

D

demon311

$F=\dfrac{ 1}{a^2+b^2+c^2}+\dfrac{ 1}{3ab}+\dfrac{ 1}{3ab}+\dfrac{ 1}{3ab} + \dfrac{ 1}{3ac}+\dfrac{ 1}{3ac}+\dfrac{ 1}{3ac}+\dfrac{ 1}{3bc}+\dfrac{ 1}{3bc}+\dfrac{ 1}{3bc} \\
\ge \dfrac{ 100}{(a+b+c)^2+7(ab+bc+ac)} \ge \dfrac{ 100}{1+\dfrac{ 7}{3}.(a+b+c)^2} = 30 \\
Min \ F =30 \leftrightarrow a=b=c=\dfrac{ 1}{3}$
 
E

eye_smile

$F \ge \dfrac{1}{a^2+b^2+c^2}+\dfrac{9}{ab+bc+ca} \ge \dfrac{9}{(a+b+c)^2}+\dfrac{7}{ab+bc+ca} \ge 30$

($ab+bc+ca \le \dfrac{1}{3}(a+b+c)^2=1/3$)
 
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