The problem asks us to find the maximum and minimum values of the function $F(x) = x^{2/3}$ on the interval $[-1, 2]$. I will go through the solution, including finding the derivative to identify critical points and then evaluating the function at these points and the endpoints.

Here is the solution in LaTeX format:

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\documentclass{article}
\usepackage{amsmath}

\begin{document}

\textbf{Problem:} \\
Find the maximum and minimum values of the function $$F(x) = x^{2/3}$$ on the interval $$[-1, 2]$$.

\textbf{Solution:} \\
1. \textit{Calculate the derivative of} $$F(x)$$: \\
   Given $$F(x) = x^{2/3}$$, we find the derivative $$F'(x)$$ to determine any critical points.

   $$F'(x) = \frac{d}{dx} \left( x^{2/3} \right) = \frac{2}{3} x^{-\frac{1}{3}} = \frac{2}{3x^{1/3}}$$

   The derivative $$F'(x)$$ is undefined at $$x = 0$$, indicating a potential critical point at $$x = 0$$.

2. \textit{Evaluate} $$F(x)$$ \textit{at the critical point and endpoints}: \\
   We need to evaluate $$F(x)$$ at $$x = -1$$, $$x = 0$$, and $$x = 2$$ to find the maximum and minimum values.

   $$F(-1) = (-1)^{2/3} = 1$$
   $$F(0) = 0^{2/3} = 0$$
   $$F(2) = 2^{2/3} = \sqrt[3]{4} \approx 1.587$$

3. \textit{Conclusion:} \\
   The maximum value of $$F(x)$$ on $$[-1, 2]$$ is approximately $$1.587$$ at $$x = 2$$, and the minimum value is $$0$$ at $$x = 0$$.

\textbf{Answer:} \\
Maximum value: $$\sqrt[3]{4} \approx 1.587$$ at $$x = 2$$ \\
Minimum value: $$0$$ at $$x = 0$$

\end{document}

Would you like further details on the calculation steps, or do you have any questions about the solution?

Here are five related questions that build on this problem:

1. How do we interpret a function's derivative when it is undefined at certain points?
2. Why is it essential to evaluate both critical points and endpoints on a closed interval?
3. How would the answer change if the interval were open instead of closed?
4. What are the implications of fractional exponents in function behavior?
5. How would you find the maximum and minimum of $F(x) = x^{2/3} + x$ on the same interval?

Tip: When working with fractional powers, always check for points where the derivative may be undefined, as they often indicate critical points.