Áp dụng bđt Cauchy Shwarz và bđt phụ \(\dfrac{1}{x+y}\le\dfrac{1}{4}\left(\dfrac{1}{x}+\dfrac{1}{y}\right)\)
\(\Rightarrow M^2=\left(\sqrt{\dfrac{a}{b+c+2a}}+\sqrt{\dfrac{b}{c+a+2b}}+\sqrt{\dfrac{c}{a+b+2c}}\right)^2\)
\(\le\left(1+1+1\right)\left(\dfrac{a}{b+c+2a}+\dfrac{b}{c+a+2b}+\dfrac{c}{a+b+2c}\right)\)
\(\le\dfrac{3}{4}\left(\dfrac{a}{b+a}+\dfrac{a}{c+a}+\dfrac{b}{b+c}+\dfrac{b}{b+a}+\dfrac{c}{c+a}+\dfrac{c}{c+b}\right)\)
\(=\dfrac{3}{4}\left(\dfrac{a+b}{a+b}+\dfrac{b+c}{b+c}+\dfrac{c+a}{c+a}\right)=\dfrac{9}{4}\)
➤ \(M\le\dfrac{3}{2}\)
Dấu "=" xảy ra ⇔ a = b = c
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