For $a,b,c \in R$ and $a,b,c>0$. Minimize $A=a^3+b^3+c^3$

Question

For $a,b,c \in R$ and $a,b,c>0$ satisfy $a^2+b^2+c^2=27$, minimize $$A=a^3+b^3+c^3$$

Answer 1

By Power-Mean Inequality,

$\left(\dfrac{a^3+b^3+c^3}{3}\right)^{1/3}\ge \left(\dfrac{a^2+b^2+c^2}{3}\right)^{1/2}=3$

$a^3+b^3+c^3 \ge 81$