ta có: \(A=\dfrac{x^2}{\left(x^2+1\right)^3}=\dfrac{x^2}{x^6+3x^4+3x^2+1}=\dfrac{1}{x^4+3x^2+3+\dfrac{1}{x^2}}\)
đặt \(x^2=a\left(a\ge0\right)\Rightarrow A=\dfrac{1}{a^2+3a+3+\dfrac{1}{a}}\)
ta đi tìm min của \(P=a^2+3a+3+\dfrac{1}{a}=a^2-a+4a+\dfrac{1}{a}+3\)
\(=\left(a^2-a+\dfrac{1}{4}\right)+\left(4a+\dfrac{1}{a}\right)+\dfrac{11}{4}=\left(a-\dfrac{1}{2}\right)^2+\left(4a+\dfrac{1}{a}\right)+\dfrac{11}{4}\)
a >0;Áp dụng BĐT cauchy: \(4a+\dfrac{1}{a}\ge2\sqrt{4a.\dfrac{1}{a}}=4\)
do đó \(P\ge4+\dfrac{11}{4}=\dfrac{27}{4}\)( vì \(\left(a-\dfrac{1}{2}\right)^2\ge0\))
\(\Rightarrow A\le\dfrac{4}{27}\)
dấu = xảy ra khi \(4a=\dfrac{1}{a}\Leftrightarrow a=\dfrac{1}{2}\left(a\ge0\right)\)và nó cũng trùng với \(\left(a-\dfrac{1}{2}\right)^2\ge0\)
khi đó \(x=\pm\dfrac{\sqrt{2}}{2}\)