Minkowski Inequality

Algebra Level 5

$\large \sqrt[3]{a^3+\dfrac{1}{b^3}}+\sqrt[3]{b^3+\dfrac{1}{c^3}}+\sqrt[3]{c^3+\dfrac{1}{a^3}}$

Given that $a,b,c$ are positive reals satisfying $a+b+c\le \frac{3}{2}$ . Determine the minimum value of the expression above to 3 decimal places.

A solution using Minkowski Inequality will be very much appreciated!