Gt$\Leftrightarrow$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=7$Áp dụng BĐT Cauchy&Bunhiacopxki:
$8a^{2}+\frac{1}{2a^{2}}\geq4$
$54b^{3}+54b^{3}+\frac{2}{9b^{2}}+\frac{2}{9b^{2}}+\frac{2}{9b^{2}}\geq10$
$16c^{4}+\frac{1}{4c^{2}}+\frac{1}{4c^{2}}\geq3$
$\frac{1}{2a^{2}}+\frac{1}{3b^{2}}+\frac{1}{2c^{2}}\geq \frac{(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^{2}}{2+3+2}=7$
$\Rightarrow S\geq4+10+3+7=24$
Dấu''='' xra$\Leftrightarrow$ a=c=$\frac{1}{2}$&b=$\frac{1}{3}$