the positive numbers

Using Hölder's inequality , we have:

$\begin{aligned} \left(\frac 1a + \frac 1b + \frac 1c \right)(a+b+c)(1+1+1) & \ge (1+1+1)^3 \\ \left(\frac 1a + \frac 1b + \frac 1c \right)(1)(3) & \ge 27 \\ \implies \frac 1a + \frac 1b + \frac 1c & \ge \boxed 9 \end{aligned}$

Equality occurs when $a=b=c = \frac 13$ .

A. M. -H. M. Inequality :

$(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 9$

$\implies \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \boxed 9$ .