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Trung sĩ
Cho các số thực dương $a,b,c$ thoả mãn $a+b+c=3$. Tìm giá trị lớn nhất của:
$$P=\frac{bc}{\sqrt[4]{a^2+3}}+\frac{ca}{\sqrt[4]{b^2+3}}+\frac{ab}{\sqrt[4]{c^2+3}}$$
Edited by DBS, 15-07-2021 - 18:47.
Thiếu úy
Ta có $\sum\frac{bc}{\sqrt[4]{a^2+3}}\leq \sum\frac{bc}{\sqrt[4]{(a+b)(a+c)}}\leq _{AM-GM}\sum\frac{1}{2\sqrt{2}}\left ( \frac{bc}{2}+\frac{bc}{2} +\frac{bc}{a+b}+\frac{bc}{a+c} \right)\leq \frac{3}{\sqrt{2}}$.
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