Cho a, b, c > 0. Tìm Max:
$A=\frac{bc}{a^{2}+3b^{2}+2c^{2}}+\frac{ca}{b^{2}+3c^{2}+2a^{2}}+\frac{ab}{c^{2}+3a^{2}+2b^{2}}$
Cho a, b, c > 0. Tìm Max:
$A=\frac{bc}{a^{2}+3b^{2}+2c^{2}}+\frac{ca}{b^{2}+3c^{2}+2a^{2}}+\frac{ab}{c^{2}+3a^{2}+2b^{2}}$
Cho a, b, c > 0. Tìm Max:
$A=\frac{bc}{a^{2}+3b^{2}+2c^{2}}+\frac{ca}{b^{2}+3c^{2}+2a^{2}}+\frac{ab}{c^{2}+3a^{2}+2b^{2}}$
Ta có :
$\frac{bc}{a^{2}+3b^{2}+2c^{2}}\leq \frac{bc}{2ab+4bc}=\frac{1}{2}.\frac{c}{2c+a}=\frac{1}{4}\left ( 1-\frac{a}{2c+a} \right )$
$\Rightarrow A=\frac{bc}{a^{2}+3b^{2}+2c^{2}}+\frac{ca}{b^{2}+3c^{2}+2a^{2}}+\frac{ab}{c^{2}+3a^{2}+2b^{2}}\leq \frac{3}{4}-\frac{1}{4}\left ( \sum \frac{a}{a+2c} \right )=\frac{3}{4}-\frac{1}{4}\left ( \sum \frac{a^2}{a^2+2ac} \right )\leq \frac{3}{4}-\frac{1}{4}\frac{\left ( a+b+c \right )^2}{\sum a^2+2ab+2ac+2bc}=\frac{1}{2}$
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