$\displaystyle \begin{array}{{>{\displaystyle}l}} a,b,c\geqslant 0.\ CMR\ ::\ a+b+c\geqslant 3\sqrt[3]{abc} \ \\ Đặt\ x^{3} =a;y^{3} =;z^{3} =c\ ( x;y;z\geqslant 0\ ) \ \\ Cần\ chứng\ minh\ x^{3} +y^{3} +z^{3} \geqslant 3xyz\ \\ ta\ có\ :\ x^{3} +y^{3} +z^{3} -3xyz\ \\ \rightarrow ( x+y)^{3} +z^{3} -3xy( x+y+z) =0\ \\ \rightarrow ( x+y+z)\left[( x+y)^{2} -( x+y) z+z^{2} -3xy\right] \geqslant 0\ \\ \rightarrow ( x+y+z)\left[ x^{2} +y^{2} +z^{2} -xy-yz-xz\right] =0\ \\ \rightarrow ( x+y+z)\left[ 2x^{2} +2y^{2} +2z^{2} -2xy-2xz-2yz\right] =0\ \\ \rightarrow ( x+y+z)\left[( x-y)^{2} +( y-z)^{2} +( z-x)^{2}\right] \geqslant 0\ ( \ đẳng\ thức\ đúng\ ) \ \\ Dấu\ =\ xảy\ ra\ khi\ x=y=z\ hay\ a=b=c\ \end{array}$
