TITLE: the least value for :$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$
QUESTION [12 upvotes]: For every $a,b,c$ non-negative real number such that:$a+b+c=1$ how to find the least value for :
$$\frac{a}{b^3+54}+\frac{b}{c^3+54}+\frac{c}{a^3+54}$$

REPLY [3 votes]: Since you said non-negative, I am going to assume that zero is allowed in which case I found a minimum value of 0.0184826 achieved at $a=0, b=0.748545, c=0.251455$ up to their cyclic permutations.
After posting this I see that Oleg567 pointed to this solution already. Furthermore, if you want all $a,b,c$ to be strictly positive then it looks like the minimum is indeed at 0.0185045 although the values for $a,b,c$ where it is achieved are not unique. In addition to Alexander Gruber and daniel's answer, I found $a=0.530893, b=0.335498, c=0.133609$.