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!

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