Cho a+b+c=8;a,b,c>0. CMR:
\(\sqrt[4]{a^3}+\sqrt[4]{b^3}+\sqrt[4]{c^3}>2\sqrt{2}\)
\(\sqrt[4]{a^3}+\sqrt[4]{b^3}+\sqrt[4]{c^3}=\sqrt[4]{\dfrac{4a^3}{4}}+\sqrt[4]{\dfrac{4b^3}{4}}+\sqrt[4]{\dfrac{4c^3}{4}}\)
\(=\dfrac{1}{\sqrt[4]{4}}\left(\sqrt[4]{\left(a+b+c\right)a^3}+\sqrt[4]{\left(a+b+c\right)b^3}+\sqrt[4]{\left(a+b+c\right)c^3}\right)\)
\(=\dfrac{1}{\sqrt{2}}\left(\sqrt[4]{a^4+a^3\left(b+c\right)}+\sqrt[4]{b^4+b^3\left(a+c\right)}+\sqrt[4]{c^4+c^3\left(a+b\right)}\right)\)
\(>\dfrac{1}{\sqrt{2}}\left(\sqrt[4]{a^4}+\sqrt[4]{b^4}+\sqrt[4]{c^4}\right)=\dfrac{1}{\sqrt{2}}\left(a+b+c\right)=\dfrac{4}{\sqrt{2}}=2\sqrt{2}\) (đpcm)
