If $a0, b0, c0$, Find the minimum value of $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\righ

Question

If $a>0, b>0, c>0$, Find the minimum value of $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$

Professor PC Thomas Classes · Answer

By A.M-G.M Theorem,\ $\frac{a+b+c}{3} \geq(a b c)^{1 / 3}$\ or $(a+b+c) \geq 3(a b c)^{1 / 3}$\ and $\left(\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}ight) \geq\left(\frac{1}{a b c}ight)^{1 / 3}$\ or $\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}ight) \geq 3\left(\frac{1}{a b c}ight)^{1 / 3}$\ Multiplying $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}ight) \geq 9$