\(\frac{a}{b^2+c^2}=\frac{a}{1-a^2}=\frac{a^2}{a-a^3}\)
Chứng minh: \(a-a^3\le\frac{2}{3\sqrt{3}}\text{ }\left(#\right)\)
\(\left(#\right)\Leftrightarrow a^3-a+\frac{2}{3\sqrt{3}}\ge0\Leftrightarrow\left(a-\frac{1}{\sqrt{3}}\right)^2\left(x+\frac{2}{\sqrt{3}}\right)\ge0\)
Bất đẳng thức cuối đúng nên có đpcm.
\(\Rightarrow P\ge\frac{1}{\frac{2}{3\sqrt{3}}}\left(a^2+b^2+c^2\right)=\frac{3\sqrt{3}}{2}\)
Cách chứng minh khac cho \(a\left(1-a^2\right)\le\frac{2}{3\sqrt{3}}\)
Áp dụng BĐT Côsi: \(a\left(1-a^2\right)=\sqrt{\frac{1}{2}.2a^2.\left(1-a^2\right).\left(1-a^2\right)}\le\sqrt{\frac{1}{2}\left(\frac{2a^2+1-a^2+1-a^2}{3}\right)^3}=\frac{2}{3\sqrt{3}}\)