Prove that for all positive real numbers $a, b$, and $c$ , $\sqrt[3]{abc}+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge 2\sqrt3$.
When does the equality occur?
1. **Applying the Arithmetic Mean-Geometric Mean Inequality (AM-GM):**

   The AM-GM inequality states that for any non-negative real numbers \( x_1, x_2, \ldots, x_n \),
   \[
   \frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}
   \]
   with equality if and only if \( x_1 = x_2 = \cdots = x_n \).

2. **Using AM-GM on the terms \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \):**

   Applying AM-GM to the terms \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \), we get:
   \[
   \frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{3} \geq \sqrt[3]{\frac{1}{a} \cdot \frac{1}{b} \cdot \frac{1}{c}}
   \]
   Simplifying the right-hand side, we have:
   \[
   \frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{3} \geq \sqrt[3]{\frac{1}{abc}}
   \]
   Therefore,
   \[
   \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 3 \sqrt[3]{\frac{1}{abc}}
   \]

3. **Combining the inequalities:**

   Let \( x = \sqrt[3]{abc} \). Then, the inequality becomes:
   \[
   \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq \frac{3}{x}
   \]
   Adding \( x \) to both sides, we get:
   \[
   x + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq x + \frac{3}{x}
   \]

4. **Analyzing the function \( f(x) = x + \frac{3}{x} \):**

   To find the minimum value of \( f(x) = x + \frac{3}{x} \), we take the derivative and set it to zero:
   \[
   f'(x) = 1 - \frac{3}{x^2}
   \]
   Setting \( f'(x) = 0 \), we get:
   \[
   1 - \frac{3}{x^2} = 0 \implies x^2 = 3 \implies x = \sqrt{3}
   \]
   Evaluating \( f(x) \) at \( x = \sqrt{3} \):
   \[
   f(\sqrt{3}) = \sqrt{3} + \frac{3}{\sqrt{3}} = \sqrt{3} + \sqrt{3} = 2\sqrt{3}
   \]

5. **Conclusion:**

   Therefore, we have shown that:
   \[
   \sqrt[3]{abc} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 2\sqrt{3}
   \]
   Equality holds when \( \sqrt[3]{abc} = \sqrt{3} \), which implies \( abc = 3\sqrt{3} \) and \( \frac{1}{a} = \frac{1}{b} = \frac{1}{c} \), or equivalently \( a = b = c = \sqrt[3]{3\sqrt{3}} \).

The final answer is \( \boxed{ \sqrt[3]{abc} + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \geq 2\sqrt{3} } \)