Áp dụng BĐT Cauchy ta có:
$4a^2+9b^2\ge12ab$
$3b^3+12c^2\ge12bc$
$2a^2+18c^2\ge12ac$
Suy ra: $6P\ge12(ab+bc+ca)=12 \Rightarrow P\ge 2$
$\min P=2 \Leftrightarrow \left\{\begin{array}{l}2a=3b=6c\\ab+bc+ca=1\end{array}\right. \Leftrightarrow \left\{\begin{array}{l}a=\dfrac{3}{\sqrt{11}}\\b=\dfrac{2}{\sqrt{11}}\\c=\dfrac{1}{\sqrt{11}}\end{array}\right.$