Ta có: \(2\left(a^5+b^5\right)=\left(a+b\right)\left(a^5+b^5\right)\ge\left(a^3+b^3\right)^2\)
\(\Rightarrow a^5+b^5\ge\frac{\left(a^3+b^3\right)^2}{2}\)
Mà \(2\left(a^3+b^3\right)=\left(a+b\right)\left(a^3+b^3\right)\ge\left(a^2+b^2\right)^2\)
\(\Rightarrow a^5+b^5\ge\frac{\left(\frac{\left(a^2+b^2\right)^2}{2}\right)^2}{2}=\frac{\left(a^2+b^2\right)^4}{8}\)
\(\ge\frac{\left(\frac{\left(a+b\right)^2}{2}\right)^4}{8}=\frac{16}{8}=2\left(đpcm\right)\)
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