TITLE: An easy inequality?
QUESTION [5 upvotes]: While solving a problem, I have struck over this inequality, as
If $a^3+b^3+c^3=15$, find minimum value of 
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$$
Can anybody help me?

REPLY [9 votes]: By Holder's Inequality:$$\left(\frac1a+\frac1b+\frac1c \right)^3(a^3+b^3+c^3) \ge (1+1+1)^4$$
$$\implies \frac1a+\frac1b+\frac1c  \ge \frac3{\sqrt[3]5}$$
with equality when $a=b=c=\sqrt[3]5$.