Lời giải:
Áp dụng BĐT AM-GM:
$4abc+4abc+\frac{1}{8a^2}+\frac{1}{8b^2}+\frac{1}{8c^2}\geq 5\sqrt[5]{\frac{1}{32}}=\frac{5}{2}(1)$
Áp dụng BĐT Cauchy_Schwarz:
$\frac{7}{8}\left(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\right)\geq \frac{7}{8}.\frac{9}{a^2+b^2+c^2}\geq \frac{7}{8}.\frac{9}{\frac{3}{4}}=\frac{21}{2}(2)$
Từ $(1);(2)\Rightarrow P\geq 13$
Vậy $P_{\min}=13$ khi $a=b=c=\frac{1}{2}$