ÁD BĐT Cauchy:$\frac{1}{a^{2}+b^{2}}+\frac{4}{\sqrt{a}}+\frac{4}{\sqrt{a}}+\frac{4}{\sqrt{b}}+\frac{4}{\sqrt{b}}\geq5\sqrt[5]{\frac{2^{11}}{4(a^{2}+b^{2})2ab}}\geq5\sqrt[5]{\frac{2^{11}}{(a+b)^{4}}}=40$(1)
$\frac{2}{\sqrt{a}}+\frac{2}{\sqrt{b}}\geq \frac{4\sqrt{2}}{\sqrt{a+b}}=8$(2)
Từ(1)&(2)$\Rightarrow đpcm$
$Dấu''='' xra\Leftrightarrow a=b=\frac{1}{4}$