Minimum value of $\frac{a^3}{b}+2\frac{b^2}{c^2}+\frac{c}{2a^3}$

Find the minimum value of $\dfrac{a^3}{b}+2\dfrac{b^2}{c^2}+\dfrac{c}{2a^3}$ where $a,b$ and $c$ are positive real numbers.

I tried to used the property $a^2+b^2 \geq 2ab$ but I couldn't figure out how to cancel these $a,b$ and $c$. Can someone help me?

• What is the context of this question? Is it meant to be a multivariable calculus problem? – Alex S Apr 2 '16 at 5:37

write $\dfrac{a^3}{2b}+\dfrac{a^3}{2b}+\dfrac{2b^2}{c^2}+\dfrac{c}{4a^3}+\dfrac{c}{4a^3}$. Then use the AM-GM inequality for 6 numbers.