Áp dụng BĐT MInkowski ta có:$P\ge\sqrt{(a+b+c)^2+(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2}\ge \sqrt{(a+b+c)^2+\frac{81}{(a+b+c)^2}}$Xét $f(t)=t^2+\frac{81}{t^2};0<t\le 2$$f'(t)=t-\frac{81}{t^3}<0 ;\forall 0<t\le 2$$\Rightarrow f(t)$ nghịch biến$\Rightarrow f(t)\ge f(2)=\frac{97}{4}$ $\Rightarrow P\ge \frac{\sqrt{97}}{2}$
Áp dụng BĐT MInkowski ta có:$P\ge\sqrt{(a+b+c)^2+(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2}\ge \sqrt{(a+b+c)^2+\frac{81}{(a+b+c)^2}}$Xét $f(t)=t^2+\frac{81}{t^2};0$f'(t)=t-\frac{81}{t^3}<0 ;\forall 0$\Rightarrow f(t)$ nghịch biến$\Rightarrow f(t)\ge f(2)=\frac{97}{4}$$\Rightarrow P\ge \frac{\sqrt{97}}{2}$
Áp dụng BĐT MInkowski ta có:$P\ge\sqrt{(a+b+c)^2+(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})^2}\ge \sqrt{(a+b+c)^2+\frac{81}{(a+b+c)^2}}$Xét $f(t)=t^2+\frac{81}{t^2};0
<t\le 2$$f'(t)=t-\frac{81}{t^3}<0 ;\forall 0
<t\le 2$$\Rightarrow f(t)$ nghịch biến$\Rightarrow f(t)\ge f(2)=\frac{97}{4}$
$\Rightarrow P\ge \frac{\sqrt{97}}{2}$