Ta có
$2(a^5+b^5)=(a^5+b^5)(a+b)\geq (a^3+b^3)^2$
$(a^5+b^5)\geq \frac{(a^3+b^3)^2}{2}$
Mà $2(a^3+b^3)=(a+b)(a^3+b^3)\geq (a^2+b^2)^2$
$\Rightarrow a^3+b^3\geq \frac{(a^2+b^2)^2}{2}$
$\Rightarrow a^5+b^5\geq\frac{( \frac{(a^2+b^2)^2}{2})^2}{2}=\frac{(a^2+b^2)^4}{8}\geq \frac{(\frac{(a+b)^2}{2})^4}{8} =\frac{(\frac{(2)^2}{2})^4}{8}=2$
Dấu bằng xảy ra khi a=b=1